Theorem bnj1529 35063

Description: Technical lemma for bnj1522 35065. 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 1912	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
3		bnj1529.2	. . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
4	3	nfcii 2892	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6917	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥𝐺
8		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥 pred(𝑦, 𝐴, 𝑅)
9	4, 8	nfres 6002	. . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅))
10	5, 9	nfop 4894	. . . . 5 ⊢ Ⅎ𝑥⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩
11	7, 10	nffv 6917	. . . 4 ⊢ Ⅎ𝑥(𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
12	6, 11	nfeq 2917	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
13		fveq2 6907	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
14		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
15		bnj602 34908	. . . . . . 7 ⊢ (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅))
16	15	reseq2d 6000	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)))
17	14, 16	opeq12d 4886	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
18	17	fveq2d 6911	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
19	13, 18	eqeq12d 2751	. . 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 3304	. 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 218	1 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))

Syntax hints:   → wi 4  ∀wal 1535   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ⟨cop 4637   ↾ cres 5691  ‘cfv 6563   predc-bnj14 34681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701  df-iota 6516  df-fv 6571  df-bnj14 34682

This theorem is referenced by:  bnj1523  35064