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Metamath Proof Explorer

Theorem bnj906

Description: Property of _trCl . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Assertion bnj906 ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) → pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) )

Proof

Step Hyp Ref Expression
1 1onn ⊢ 1o ∈ ω
2 1n0 ⊢ 1o ≠ ∅
3 eldifsn ⊢ ( 1o ∈ ( ω ∖ { ∅ } ) ↔ ( 1o ∈ ω ∧ 1o ≠ ∅ ) )
4 1 2 3 mpbir2an ⊢ 1o ∈ ( ω ∖ { ∅ } )
5 4 ne0ii ⊢ ( ω ∖ { ∅ } ) ≠ ∅
6 biid ⊢ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
7 biid ⊢ ( ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
8 eqid ⊢ ( ω ∖ { ∅ } ) = ( ω ∖ { ∅ } )
9 6 7 8 bnj852 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∀ 𝑛 ∈ ( ω ∖ { ∅ } ) ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
10 r19.2z ⊢ ( ( ( ω ∖ { ∅ } ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( ω ∖ { ∅ } ) ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) → ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
11 5 9 10 sylancr ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
12 euex ⊢ ( ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) → ∃ 𝑓 ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
13 11 12 bnj31 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ∃ 𝑓 ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
14 rexcom4 ⊢ ( ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ∃ 𝑓 ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ↔ ∃ 𝑓 ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
15 13 14 sylib ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃ 𝑓 ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
16 abid ⊢ ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ↔ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
17 15 16 bnj1198 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃ 𝑓 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } )
18 simp2 ⊢ ( ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) → ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
19 18 reximi ⊢ ( ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) → ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
20 16 19 sylbi ⊢ ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
21 df-rex ⊢ ( ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ∃ 𝑛 ( 𝑛 ∈ ( ω ∖ { ∅ } ) ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) )
22 19.41v ⊢ ( ∃ 𝑛 ( 𝑛 ∈ ( ω ∖ { ∅ } ) ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) ↔ ( ∃ 𝑛 𝑛 ∈ ( ω ∖ { ∅ } ) ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) )
23 22 simprbi ⊢ ( ∃ 𝑛 ( 𝑛 ∈ ( ω ∖ { ∅ } ) ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) → ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
24 21 23 sylbi ⊢ ( ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) → ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
25 20 24 syl ⊢ ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
26 eqid ⊢ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } = { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) }
27 8 26 bnj900 ⊢ ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → ∅ ∈ dom 𝑓 )
28 fveq2 ⊢ ( 𝑖 = ∅ → ( 𝑓 ‘ 𝑖 ) = ( 𝑓 ‘ ∅ ) )
29 28 ssiun2s ⊢ ( ∅ ∈ dom 𝑓 → ( 𝑓 ‘ ∅ ) ⊆ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) )
30 27 29 syl ⊢ ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → ( 𝑓 ‘ ∅ ) ⊆ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) )
31 ssiun2 ⊢ ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) ⊆ ∪ 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) )
32 6 7 8 26 bnj882 ⊢ trCl ( 𝑋 , 𝐴 , 𝑅 ) = ∪ 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 )
33 31 32 sseqtrrdi ⊢ ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) )
34 30 33 sstrd ⊢ ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → ( 𝑓 ‘ ∅ ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) )
35 25 34 eqsstrrd ⊢ ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) )
36 35 exlimiv ⊢ ( ∃ 𝑓 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) )
37 17 36 syl ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) → pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) )