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Theorem bnj1520 31603
Description: Technical lemma for bnj1500 31605. 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 31361 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
43nfcii 2898 . . . . . 6 𝑓𝐶
54nfuni 4602 . . . . 5 𝑓 𝐶
61, 5nfcxfr 2905 . . . 4 𝑓𝐹
7 nfcv 2907 . . . 4 𝑓𝑥
86, 7nffv 6389 . . 3 𝑓(𝐹𝑥)
9 nfcv 2907 . . . 4 𝑓𝐺
10 nfcv 2907 . . . . . 6 𝑓 pred(𝑥, 𝐴, 𝑅)
116, 10nfres 5569 . . . . 5 𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
127, 11nfop 4577 . . . 4 𝑓𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
139, 12nffv 6389 . . 3 𝑓(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
148, 13nfeq 2919 . 2 𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
1514nf5ri 2227 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 1650   = wceq 1652  {cab 2751  wral 3055  wrex 3056  wss 3734  cop 4342   cuni 4596  cres 5281   Fn wfn 6065  cfv 6070   predc-bnj14 31226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-xp 5285  df-res 5291  df-iota 6033  df-fv 6078
This theorem is referenced by:  bnj1501  31604
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