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Metamath Proof Explorer

Theorem bnj579

Description: Technical lemma for bnj852 . 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 bnj579.1 ⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
bnj579.2 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj579.3 ⊢ 𝐷 = ( ω ∖ { ∅ } )
Assertion bnj579 ( 𝑛 ∈ 𝐷 → ∃* 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 bnj579.1 ⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
2 bnj579.2 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3 bnj579.3 ⊢ 𝐷 = ( ω ∖ { ∅ } )
4 biid ⊢ ( ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
5 biid ⊢ ( [ 𝑔 / 𝑓 ] 𝜑 ↔ [ 𝑔 / 𝑓 ] 𝜑 )
6 biid ⊢ ( [ 𝑔 / 𝑓 ] 𝜓 ↔ [ 𝑔 / 𝑓 ] 𝜓 )
7 biid ⊢ ( [ 𝑔 / 𝑓 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ [ 𝑔 / 𝑓 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
8 biid ⊢ ( ( ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ [ 𝑔 / 𝑓 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ) ↔ ( ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ [ 𝑔 / 𝑓 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ) )
9 biid ⊢ ( ∀ 𝑘 ∈ 𝑛 ( 𝑘 E 𝑗 → [ 𝑘 / 𝑗 ] ( ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ [ 𝑔 / 𝑓 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ) ) ↔ ∀ 𝑘 ∈ 𝑛 ( 𝑘 E 𝑗 → [ 𝑘 / 𝑗 ] ( ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ [ 𝑔 / 𝑓 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ) ) )
10 1 2 4 5 6 7 3 8 9 bnj580 ⊢ ( 𝑛 ∈ 𝐷 → ∃* 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )