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Theorem bnj1519 31440
Description: Technical lemma for bnj1500 31443. 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 3143 . . . . . . . 8 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
43nfab 2907 . . . . . . 7 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
52, 4nfcxfr 2900 . . . . . 6 𝑑𝐶
65nfuni 4594 . . . . 5 𝑑 𝐶
71, 6nfcxfr 2900 . . . 4 𝑑𝐹
8 nfcv 2902 . . . 4 𝑑𝑥
97, 8nffv 6359 . . 3 𝑑(𝐹𝑥)
10 nfcv 2902 . . . 4 𝑑𝐺
11 nfcv 2902 . . . . . 6 𝑑 pred(𝑥, 𝐴, 𝑅)
127, 11nfres 5553 . . . . 5 𝑑(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
138, 12nfop 4569 . . . 4 𝑑𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
1410, 13nffv 6359 . . 3 𝑑(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
159, 14nfeq 2914 . 2 𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
1615nf5ri 2212 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 383  wal 1630   = wceq 1632  {cab 2746  wral 3050  wrex 3051  wss 3715  cop 4327   cuni 4588  cres 5268   Fn wfn 6044  cfv 6049   predc-bnj14 31063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-res 5278  df-iota 6012  df-fv 6057
This theorem is referenced by:  bnj1501  31442
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