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Metamath Proof Explorer

Theorem bnj882

Description: Definition (using hypotheses for readability) of the function giving the transitive closure of X in A by R . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj882.1 ⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
bnj882.2 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj882.3 ⊢ 𝐷 = ( ω ∖ { ∅ } )
bnj882.4 ⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
Assertion bnj882 trCl ( 𝑋 , 𝐴 , 𝑅 ) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 )

Proof

Step Hyp Ref Expression
1 bnj882.1 ⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2 bnj882.2 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3 bnj882.3 ⊢ 𝐷 = ( ω ∖ { ∅ } )
4 bnj882.4 ⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
5 df-bnj18 ⊢ 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 )
6 df-iun ⊢ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) = { 𝑤 ∣ ∃ 𝑓 ∈ 𝐵 𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) }
7 df-iun ⊢ ∪ 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 ) = { 𝑤 ∣ ∃ 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 ) ) ) } 𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) }
8 1 2 anbi12i ⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
9 8 anbi2i ⊢ ( ( 𝑓 Fn 𝑛 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝑓 Fn 𝑛 ∧ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
10 3anass ⊢ ( ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑓 Fn 𝑛 ∧ ( 𝜑 ∧ 𝜓 ) ) )
11 3anass ⊢ ( ( 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 ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
12 9 10 11 3bitr4i ⊢ ( ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
13 3 12 rexeqbii ⊢ ( ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ∃ 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 13 abbii ⊢ { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) } = { 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 4 14 eqtri ⊢ 𝐵 = { 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 15 eleq2i ⊢ ( 𝑓 ∈ 𝐵 ↔ 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 ) ) ) } )
17 16 anbi1i ⊢ ( ( 𝑓 ∈ 𝐵 ∧ 𝑤 ∈ ∪ 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 ) ) ) } ∧ 𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) ) )
18 17 rexbii2 ⊢ ( ∃ 𝑓 ∈ 𝐵 𝑤 ∈ ∪ 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 ) ) ) } 𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) )
19 18 abbii ⊢ { 𝑤 ∣ ∃ 𝑓 ∈ 𝐵 𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) } = { 𝑤 ∣ ∃ 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 ) ) ) } 𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) }
20 7 19 eqtr4i ⊢ ∪ 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 ) = { 𝑤 ∣ ∃ 𝑓 ∈ 𝐵 𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) }
21 6 20 eqtr4i ⊢ ∪ 𝑓 ∈ 𝐵 ∪ 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 )
22 5 21 eqtr4i ⊢ trCl ( 𝑋 , 𝐴 , 𝑅 ) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 )