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Theorem bnj1520 31433
 Description: Technical lemma for bnj1500 31435. 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 31191 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
43nfcii 2885 . . . . . 6 𝑓𝐶
54nfuni 4586 . . . . 5 𝑓 𝐶
61, 5nfcxfr 2892 . . . 4 𝑓𝐹
7 nfcv 2894 . . . 4 𝑓𝑥
86, 7nffv 6351 . . 3 𝑓(𝐹𝑥)
9 nfcv 2894 . . . 4 𝑓𝐺
10 nfcv 2894 . . . . . 6 𝑓 pred(𝑥, 𝐴, 𝑅)
116, 10nfres 5545 . . . . 5 𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
127, 11nfop 4561 . . . 4 𝑓𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
139, 12nffv 6351 . . 3 𝑓(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
148, 13nfeq 2906 . 2 𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
1514nf5ri 2204 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 383  ∀wal 1622   = wceq 1624  {cab 2738  ∀wral 3042  ∃wrex 3043   ⊆ wss 3707  ⟨cop 4319  ∪ cuni 4580   ↾ cres 5260   Fn wfn 6036  ‘cfv 6041   predc-bnj14 31055 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-xp 5264  df-res 5270  df-iota 6004  df-fv 6049 This theorem is referenced by:  bnj1501  31434
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