Theorem bnj1529 35767

Description: Technical lemma for bnj1522 35769. 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.)

Hypotheses

Ref	Expression
bnj1529.1	⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
bnj1529.2	⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)

Assertion

Ref	Expression
bnj1529	⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))

Distinct variable groups:   𝑤,𝐴,𝑥,𝑦   𝑤,𝐹,𝑦   𝑤,𝐺,𝑥,𝑦   𝑤,𝑅,𝑥,𝑦

Allowed substitution hints:   𝜒(𝑥, 𝑦, 𝑤)   𝐹(𝑥)

Proof of Theorem bnj1529

Step	Hyp	Ref	Expression
1		bnj1529.1	. 2 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
2		nfv 1949	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
3		bnj1529.2	. . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
4	3	nfcii 2915	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6915	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝐺
8		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥 pred(𝑦, 𝐴, 𝑅)
9	4, 8	nfres 5982	. . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅))
10	5, 9	nfop 4852	. . . . 5 ⊢ Ⅎ𝑥⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩
11	7, 10	nffv 6915	. . . 4 ⊢ Ⅎ𝑥(𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
12	6, 11	nfeq 2939	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
13		fveq2 6905	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
14		id 23	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
15		bnj602 35612	. . . . . . 7 ⊢ (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅))
16	15	reseq2d 5980	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)))
17	14, 16	opeq12d 4851	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
18	17	fveq2d 6909	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
19	13, 18	eqeq12d 2779	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)))
20	2, 12, 19	cbvralw 3308	. 2 ⊢ (∀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 221	1 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⟨cop 4838   ↾ cres 5657  ‘cfv 6555   predc-bnj14 35386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-res 5667  df-iota 6509  df-fv 6563  df-bnj14 35387

This theorem is used by:  bnj1523  35768