Metamath Proof Explorer

Theorem bnj938

Description: Technical lemma for bnj69 . 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
Hypotheses bnj938.1 ⊢ 𝐷 = ( ω ∖ { ∅ } )
bnj938.2 ⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
bnj938.3 ⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
bnj938.4 ⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
bnj938.5 ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion bnj938 ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )

Proof

Step Hyp Ref Expression
1 bnj938.1 ⊢ 𝐷 = ( ω ∖ { ∅ } )
2 bnj938.2 ⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
3 bnj938.3 ⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
4 bnj938.4 ⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
5 bnj938.5 ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
6 elisset ⊢ ( 𝑋 ∈ 𝐴 → ∃ 𝑥 𝑥 = 𝑋 )
7 6 bnj706 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∃ 𝑥 𝑥 = 𝑋 )
8 bnj291 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) ∧ 𝑋 ∈ 𝐴 ) )
9 8 simplbi ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) )
10 bnj602 ⊢ ( 𝑥 = 𝑋 → pred ( 𝑥 , 𝐴 , 𝑅 ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
11 10 eqeq2d ⊢ ( 𝑥 = 𝑋 → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) )
12 11 4 bitr4di ⊢ ( 𝑥 = 𝑋 → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ 𝜑′ ) )
13 12 3anbi2d ⊢ ( 𝑥 = 𝑋 → ( ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) ) )
14 13 2 bitr4di ⊢ ( 𝑥 = 𝑋 → ( ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ↔ 𝜏 ) )
15 14 3anbi2d ⊢ ( 𝑥 = 𝑋 → ( ( 𝑅 FrSe 𝐴 ∧ ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ∧ 𝜎 ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) ) )
16 9 15 syl5ibr ⊢ ( 𝑥 = 𝑋 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ( 𝑅 FrSe 𝐴 ∧ ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ∧ 𝜎 ) ) )
17 biid ⊢ ( ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ↔ ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) )
18 biid ⊢ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
19 1 17 3 18 5 bnj546 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )
20 16 19 syl6 ⊢ ( 𝑥 = 𝑋 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V ) )
21 20 exlimiv ⊢ ( ∃ 𝑥 𝑥 = 𝑋 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V ) )
22 7 21 mpcom ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )