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Theorem bnj1520 30842
Description: Technical lemma for bnj1500 30844. 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
bnj1520.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1520.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1520.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1520.4 𝐹 = 𝐶
Assertion
Ref Expression
bnj1520 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑅,𝑓   𝑥,𝑓
Allowed substitution hints:   𝐴(𝑥,𝑑)   𝐵(𝑥,𝑓,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝑅(𝑥,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝐺(𝑥,𝑑)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj1520
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1520.4 . . . . 5 𝐹 = 𝐶
2 bnj1520.3 . . . . . . . 8 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
32bnj1317 30600 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
43nfcii 2752 . . . . . 6 𝑓𝐶
54nfuni 4408 . . . . 5 𝑓 𝐶
61, 5nfcxfr 2759 . . . 4 𝑓𝐹
7 nfcv 2761 . . . 4 𝑓𝑥
86, 7nffv 6155 . . 3 𝑓(𝐹𝑥)
9 nfcv 2761 . . . 4 𝑓𝐺
10 nfcv 2761 . . . . . 6 𝑓 pred(𝑥, 𝐴, 𝑅)
116, 10nfres 5358 . . . . 5 𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
127, 11nfop 4386 . . . 4 𝑓𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
139, 12nffv 6155 . . 3 𝑓(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
148, 13nfeq 2772 . 2 𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
1514nf5ri 2063 1 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  {cab 2607  wral 2907  wrex 2908  wss 3555  cop 4154   cuni 4402  cres 5076   Fn wfn 5842  cfv 5847   predc-bnj14 30461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-res 5086  df-iota 5810  df-fv 5855
This theorem is referenced by:  bnj1501  30843
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