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Theorem bnj1520 32235
Description: Technical lemma for bnj1500 32237. 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 31992 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
43nfcii 2962 . . . . . 6 𝑓𝐶
54nfuni 4837 . . . . 5 𝑓 𝐶
61, 5nfcxfr 2972 . . . 4 𝑓𝐹
7 nfcv 2974 . . . 4 𝑓𝑥
86, 7nffv 6673 . . 3 𝑓(𝐹𝑥)
9 nfcv 2974 . . . 4 𝑓𝐺
10 nfcv 2974 . . . . . 6 𝑓 pred(𝑥, 𝐴, 𝑅)
116, 10nfres 5848 . . . . 5 𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
127, 11nfop 4811 . . . 4 𝑓𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
139, 12nffv 6673 . . 3 𝑓(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
148, 13nfeq 2988 . 2 𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
1514nf5ri 2185 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 396  wal 1526   = wceq 1528  {cab 2796  wral 3135  wrex 3136  wss 3933  cop 4563   cuni 4830  cres 5550   Fn wfn 6343  cfv 6348   predc-bnj14 31857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-res 5560  df-iota 6307  df-fv 6356
This theorem is referenced by:  bnj1501  32236
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