Theorem bnj1529 35467

Description: Technical lemma for bnj1522 35469. 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 1943	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
3		bnj1529.2	. . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
4	3	nfcii 2913	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6891	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑥𝐺
8		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑥 pred(𝑦, 𝐴, 𝑅)
9	4, 8	nfres 5979	. . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅))
10	5, 9	nfop 4853	. . . . 5 ⊢ Ⅎ𝑥⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩
11	7, 10	nffv 6891	. . . 4 ⊢ Ⅎ𝑥(𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
12	6, 11	nfeq 2937	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
13		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
14		id 23	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
15		bnj602 35312	. . . . . . 7 ⊢ (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅))
16	15	reseq2d 5977	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)))
17	14, 16	opeq12d 4845	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
18	17	fveq2d 6885	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
19	13, 18	eqeq12d 2778	. . 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 3306	. 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 1567   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ⟨cop 4594   ↾ cres 5662  ‘cfv 6536   predc-bnj14 35086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5666  df-res 5672  df-iota 6492  df-fv 6544  df-bnj14 35087

This theorem is used by:  bnj1523  35468