Theorem bnj1529

Description: Technical lemma for bnj1522 . 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	bnj1529.1	⊢ ( 𝜒 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
	bnj1529.2	⊢ ( 𝑤 ∈ 𝐹 → ∀ 𝑥 𝑤 ∈ 𝐹 )
Assertion	bnj1529	⊢ ( 𝜒 → ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1529.1	⊢ ( 𝜒 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
2		bnj1529.2	⊢ ( 𝑤 ∈ 𝐹 → ∀ 𝑥 𝑤 ∈ 𝐹 )
3		nfv	⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ )
4	2	nfcii	⊢ Ⅎ 𝑥 𝐹
5		nfcv	⊢ Ⅎ 𝑥 𝑦
6	4 5	nffv	⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 )
7		nfcv	⊢ Ⅎ 𝑥 𝐺
8		nfcv	⊢ Ⅎ 𝑥 pred ( 𝑦 , 𝐴 , 𝑅 )
9	4 8	nfres	⊢ Ⅎ 𝑥 ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) )
10	5 9	nfop	⊢ Ⅎ 𝑥 ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩
11	7 10	nffv	⊢ Ⅎ 𝑥 ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ )
12	6 11	nfeq	⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ )
13		fveq2	⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) )
14		id	⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 )
15		bnj602	⊢ ( 𝑥 = 𝑦 → pred ( 𝑥 , 𝐴 , 𝑅 ) = pred ( 𝑦 , 𝐴 , 𝑅 ) )
16	15	reseq2d	⊢ ( 𝑥 = 𝑦 → ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) = ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
17	14 16	opeq12d	⊢ ( 𝑥 = 𝑦 → ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ )
18	17	fveq2d	⊢ ( 𝑥 = 𝑦 → ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) )
19	13 18	eqeq12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ↔ ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) ) )
20	3 12 19	cbvralw	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) )
21	1 20	sylib	⊢ ( 𝜒 → ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) )