Theorem bnj155

Description: Technical lemma for bnj153 . 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	bnj155.1	⊢ ( 𝜓1 ↔ [ 𝑔 / 𝑓 ] 𝜓′ )
	bnj155.2	⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 1o → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion	bnj155	⊢ ( 𝜓1 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 1o → ( 𝑔 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑔 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj155.1	⊢ ( 𝜓1 ↔ [ 𝑔 / 𝑓 ] 𝜓′ )
2		bnj155.2	⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 1o → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3	2	sbcbii	⊢ ( [ 𝑔 / 𝑓 ] 𝜓′ ↔ [ 𝑔 / 𝑓 ] ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 1o → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
4		vex	⊢ 𝑔 ∈ V
5		fveq1	⊢ ( 𝑓 = 𝑔 → ( 𝑓 ‘ suc 𝑖 ) = ( 𝑔 ‘ suc 𝑖 ) )
6		fveq1	⊢ ( 𝑓 = 𝑔 → ( 𝑓 ‘ 𝑖 ) = ( 𝑔 ‘ 𝑖 ) )
7	6	iuneq1d	⊢ ( 𝑓 = 𝑔 → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) = ∪ 𝑦 ∈ ( 𝑔 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
8	5 7	eqeq12d	⊢ ( 𝑓 = 𝑔 → ( ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ↔ ( 𝑔 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑔 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
9	8	imbi2d	⊢ ( 𝑓 = 𝑔 → ( ( suc 𝑖 ∈ 1o → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( suc 𝑖 ∈ 1o → ( 𝑔 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑔 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
10	9	ralbidv	⊢ ( 𝑓 = 𝑔 → ( ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 1o → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 1o → ( 𝑔 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑔 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
11	4 10	sbcie	⊢ ( [ 𝑔 / 𝑓 ] ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 1o → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 1o → ( 𝑔 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑔 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
12	1 3 11	3bitri	⊢ ( 𝜓1 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 1o → ( 𝑔 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑔 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )