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Theorem bnj1519 31907
Description: Technical lemma for bnj1500 31910. 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
bnj1519.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1519.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1519.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1519.4 𝐹 = 𝐶
Assertion
Ref Expression
bnj1519 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Distinct variable groups:   𝐴,𝑑   𝐺,𝑑   𝑅,𝑑   𝑥,𝑑
Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑥,𝑓,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝑅(𝑥,𝑓)   𝐹(𝑥,𝑓,𝑑)   𝐺(𝑥,𝑓)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj1519
StepHypRef Expression
1 bnj1519.4 . . . . 5 𝐹 = 𝐶
2 bnj1519.3 . . . . . . 7 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
3 nfre1 3266 . . . . . . . 8 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
43nfab 2953 . . . . . . 7 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
52, 4nfcxfr 2945 . . . . . 6 𝑑𝐶
65nfuni 4745 . . . . 5 𝑑 𝐶
71, 6nfcxfr 2945 . . . 4 𝑑𝐹
8 nfcv 2947 . . . 4 𝑑𝑥
97, 8nffv 6540 . . 3 𝑑(𝐹𝑥)
10 nfcv 2947 . . . 4 𝑑𝐺
11 nfcv 2947 . . . . . 6 𝑑 pred(𝑥, 𝐴, 𝑅)
127, 11nfres 5728 . . . . 5 𝑑(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
138, 12nfop 4720 . . . 4 𝑑𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
1410, 13nffv 6540 . . 3 𝑑(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
159, 14nfeq 2958 . 2 𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
1615nf5ri 2157 1 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1518   = wceq 1520  {cab 2773  wral 3103  wrex 3104  wss 3854  cop 4472   cuni 4739  cres 5437   Fn wfn 6212  cfv 6217   predc-bnj14 31531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-xp 5441  df-res 5447  df-iota 6181  df-fv 6225
This theorem is referenced by:  bnj1501  31909
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