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Theorem bnj1520 32629
Description: Technical lemma for bnj1500 32631. 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 32384 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
43nfcii 2884 . . . . . 6 𝑓𝐶
54nfuni 4813 . . . . 5 𝑓 𝐶
61, 5nfcxfr 2898 . . . 4 𝑓𝐹
7 nfcv 2900 . . . 4 𝑓𝑥
86, 7nffv 6696 . . 3 𝑓(𝐹𝑥)
9 nfcv 2900 . . . 4 𝑓𝐺
10 nfcv 2900 . . . . . 6 𝑓 pred(𝑥, 𝐴, 𝑅)
116, 10nfres 5837 . . . . 5 𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
127, 11nfop 4787 . . . 4 𝑓𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
139, 12nffv 6696 . . 3 𝑓(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
148, 13nfeq 2913 . 2 𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
1514nf5ri 2197 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 399  wal 1540   = wceq 1542  {cab 2717  wral 3054  wrex 3055  wss 3853  cop 4532   cuni 4806  cres 5537   Fn wfn 6344  cfv 6349   predc-bnj14 32249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-xp 5541  df-res 5547  df-iota 6307  df-fv 6357
This theorem is referenced by:  bnj1501  32630
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