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Theorem bnj1518 35057
Description: Technical lemma for bnj1500 35061. 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 1912 . . . 4 𝑑𝜑
3 bnj1518.3 . . . . . 6 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
4 nfre1 3283 . . . . . . 7 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
54nfab 2909 . . . . . 6 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
63, 5nfcxfr 2901 . . . . 5 𝑑𝐶
76nfcri 2895 . . . 4 𝑑 𝑓𝐶
8 nfv 1912 . . . 4 𝑑 𝑥 ∈ dom 𝑓
92, 7, 8nf3an 1899 . . 3 𝑑(𝜑𝑓𝐶𝑥 ∈ dom 𝑓)
101, 9nfxfr 1850 . 2 𝑑𝜓
1110nf5ri 2193 1 (𝜓 → ∀𝑑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963  cop 4637   cuni 4912  dom cdm 5689  cres 5691   Fn wfn 6558  cfv 6563   predc-bnj14 34681   FrSe w-bnj15 34685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069
This theorem is referenced by:  bnj1501  35060
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