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Theorem bnj1519 32565
 Description: Technical lemma for bnj1500 32568. 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   𝑓,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 3230 . . . . . . . 8 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
43nfab 2925 . . . . . . 7 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
52, 4nfcxfr 2917 . . . . . 6 𝑑𝐶
65nfuni 4805 . . . . 5 𝑑 𝐶
71, 6nfcxfr 2917 . . . 4 𝑑𝐹
8 nfcv 2919 . . . 4 𝑑𝑥
97, 8nffv 6668 . . 3 𝑑(𝐹𝑥)
10 nfcv 2919 . . . 4 𝑑𝐺
11 nfcv 2919 . . . . . 6 𝑑 pred(𝑥, 𝐴, 𝑅)
127, 11nfres 5825 . . . . 5 𝑑(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
138, 12nfop 4779 . . . 4 𝑑𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
1410, 13nffv 6668 . . 3 𝑑(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
159, 14nfeq 2932 . 2 𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
1615nf5ri 2193 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 399  ∀wal 1536   = wceq 1538  {cab 2735  ∀wral 3070  ∃wrex 3071   ⊆ wss 3858  ⟨cop 4528  ∪ cuni 4798   ↾ cres 5526   Fn wfn 6330  ‘cfv 6335   predc-bnj14 32186 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-xp 5530  df-res 5536  df-iota 6294  df-fv 6343 This theorem is referenced by:  bnj1501  32567
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