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Theorem bnj1529 31476
Description: Technical lemma for bnj1522 31478. 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 1995 . . 3 𝑦(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
3 bnj1529.2 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
43nfcii 2904 . . . . 5 𝑥𝐹
5 nfcv 2913 . . . . 5 𝑥𝑦
64, 5nffv 6339 . . . 4 𝑥(𝐹𝑦)
7 nfcv 2913 . . . . 5 𝑥𝐺
8 nfcv 2913 . . . . . . 7 𝑥 pred(𝑦, 𝐴, 𝑅)
94, 8nfres 5536 . . . . . 6 𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅))
105, 9nfop 4555 . . . . 5 𝑥𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩
117, 10nffv 6339 . . . 4 𝑥(𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
126, 11nfeq 2925 . . 3 𝑥(𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
13 fveq2 6332 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
14 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
15 bnj602 31323 . . . . . . 7 (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅))
1615reseq2d 5534 . . . . . 6 (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)))
1714, 16opeq12d 4547 . . . . 5 (𝑥 = 𝑦 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
1817fveq2d 6336 . . . 4 (𝑥 = 𝑦 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
1913, 18eqeq12d 2786 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)))
202, 12, 19cbvral 3316 . 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 1629   = wceq 1631  wcel 2145  wral 3061  cop 4322  cres 5251  cfv 6031   predc-bnj14 31094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-res 5261  df-iota 5994  df-fv 6039  df-bnj14 31095
This theorem is referenced by:  bnj1523  31477
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