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Metamath Proof Explorer

Theorem bnj1318

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Assertion bnj1318 ( 𝑋 = 𝑌 → trCl ( 𝑋 , 𝐴 , 𝑅 ) = trCl ( 𝑌 , 𝐴 , 𝑅 ) )

Proof

Step Hyp Ref Expression
1 bnj602 ⊢ ( 𝑋 = 𝑌 → pred ( 𝑋 , 𝐴 , 𝑅 ) = pred ( 𝑌 , 𝐴 , 𝑅 ) )
2 1 eqeq2d ⊢ ( 𝑋 = 𝑌 → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑌 , 𝐴 , 𝑅 ) ) )
3 2 3anbi2d ⊢ ( 𝑋 = 𝑌 → ( ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
4 3 rexbidv ⊢ ( 𝑋 = 𝑌 → ( ∃ 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 ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
5 4 abbidv ⊢ ( 𝑋 = 𝑌 → { 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 ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } )
6 hbab1 ⊢ ( 𝑧 ∈ { 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 𝑧 ∈ { 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 ) ) ) } )
7 hbab1 ⊢ ( 𝑧 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } → ∀ 𝑓 𝑧 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } )
8 6 7 bnj1316 ⊢ ( { 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 ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 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 ) = ∪ 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) )
9 5 8 syl ⊢ ( 𝑋 = 𝑌 → ∪ 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 ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) )
10 biid ⊢ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
11 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 ) ) )
12 eqid ⊢ ( ω ∖ { ∅ } ) = ( ω ∖ { ∅ } )
13 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 ) ) ) }
14 10 11 12 13 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 )
15 biid ⊢ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑌 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑌 , 𝐴 , 𝑅 ) )
16 eqid ⊢ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑌 , 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 ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) }
17 15 11 12 16 bnj882 ⊢ trCl ( 𝑌 , 𝐴 , 𝑅 ) = ∪ 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑌 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 )
18 9 14 17 3eqtr4g ⊢ ( 𝑋 = 𝑌 → trCl ( 𝑋 , 𝐴 , 𝑅 ) = trCl ( 𝑌 , 𝐴 , 𝑅 ) )