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Theorem bnj1529 34547
Description: Technical lemma for bnj1522 34549. 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 1916 . . 3 𝑦(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
3 bnj1529.2 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
43nfcii 2886 . . . . 5 𝑥𝐹
5 nfcv 2902 . . . . 5 𝑥𝑦
64, 5nffv 6901 . . . 4 𝑥(𝐹𝑦)
7 nfcv 2902 . . . . 5 𝑥𝐺
8 nfcv 2902 . . . . . . 7 𝑥 pred(𝑦, 𝐴, 𝑅)
94, 8nfres 5983 . . . . . 6 𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅))
105, 9nfop 4889 . . . . 5 𝑥𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩
117, 10nffv 6901 . . . 4 𝑥(𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
126, 11nfeq 2915 . . 3 𝑥(𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
13 fveq2 6891 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
14 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
15 bnj602 34392 . . . . . . 7 (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅))
1615reseq2d 5981 . . . . . 6 (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)))
1714, 16opeq12d 4881 . . . . 5 (𝑥 = 𝑦 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
1817fveq2d 6895 . . . 4 (𝑥 = 𝑦 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
1913, 18eqeq12d 2747 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩)))
202, 12, 19cbvralw 3302 . 2 (∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
211, 20sylib 217 1 (𝜒 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wcel 2105  wral 3060  cop 4634  cres 5678  cfv 6543   predc-bnj14 34165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-res 5688  df-iota 6495  df-fv 6551  df-bnj14 34166
This theorem is referenced by:  bnj1523  34548
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