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Theorem bnj1529 31264
 Description: Technical lemma for bnj1522 31266. 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 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
bnj1529.2 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
Assertion
Ref Expression
bnj1529 (𝜒 → ∀𝑦𝐴 (𝐹𝑦) = (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
StepHypRef Expression
1 bnj1529.1 . 2 (𝜒 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
2 nfv 1883 . . 3 𝑦(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
3 bnj1529.2 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
43nfcii 2784 . . . . 5 𝑥𝐹
5 nfcv 2793 . . . . 5 𝑥𝑦
64, 5nffv 6236 . . . 4 𝑥(𝐹𝑦)
7 nfcv 2793 . . . . 5 𝑥𝐺
8 nfcv 2793 . . . . . . 7 𝑥 pred(𝑦, 𝐴, 𝑅)
94, 8nfres 5430 . . . . . 6 𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅))
105, 9nfop 4449 . . . . 5 𝑥𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩
117, 10nffv 6236 . . . 4 𝑥(𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
126, 11nfeq 2805 . . 3 𝑥(𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
13 fveq2 6229 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
14 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
15 bnj602 31111 . . . . . . 7 (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅))
1615reseq2d 5428 . . . . . 6 (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)))
1714, 16opeq12d 4441 . . . . 5 (𝑥 = 𝑦 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
1817fveq2d 6233 . . . 4 (𝑥 = 𝑦 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
1913, 18eqeq12d 2666 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)))
202, 12, 19cbvral 3197 . 2 (∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
211, 20sylib 208 1 (𝜒 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ⟨cop 4216   ↾ cres 5145  ‘cfv 5926   predc-bnj14 30882 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-res 5155  df-iota 5889  df-fv 5934  df-bnj14 30883 This theorem is referenced by:  bnj1523  31265
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