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Theorem bnj1518 30386
 Description: Technical lemma for bnj1500 30390. 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
bnj1518.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1518.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1518.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1518.4 𝐹 = 𝐶
bnj1518.5 (𝜑 ↔ (𝑅 FrSe 𝐴𝑥𝐴))
bnj1518.6 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
Assertion
Ref Expression
bnj1518 (𝜓 → ∀𝑑𝜓)
Distinct variable groups:   𝑓,𝑑   𝜑,𝑑   𝑥,𝑑
Allowed substitution hints:   𝜑(𝑥,𝑓)   𝜓(𝑥,𝑓,𝑑)   𝐴(𝑥,𝑓,𝑑)   𝐵(𝑥,𝑓,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝑅(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝐺(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj1518
StepHypRef Expression
1 bnj1518.6 . . 3 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
2 nfv 1830 . . . 4 𝑑𝜑
3 bnj1518.3 . . . . . 6 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
4 nfre1 2988 . . . . . . 7 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
54nfab 2755 . . . . . 6 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
63, 5nfcxfr 2749 . . . . 5 𝑑𝐶
76nfcri 2745 . . . 4 𝑑 𝑓𝐶
8 nfv 1830 . . . 4 𝑑 𝑥 ∈ dom 𝑓
92, 7, 8nf3an 1819 . . 3 𝑑(𝜑𝑓𝐶𝑥 ∈ dom 𝑓)
101, 9nfxfr 1771 . 2 𝑑𝜓
1110nf5ri 2053 1 (𝜓 → ∀𝑑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ⟨cop 4131  ∪ cuni 4372  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804   predc-bnj14 30007   FrSe w-bnj15 30011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902 This theorem is referenced by:  bnj1501  30389
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