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Theorem bnj1518 31452
Description: Technical lemma for bnj1500 31456. 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 2008 . . . 4 𝑑𝜑
3 bnj1518.3 . . . . . 6 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
4 nfre1 3191 . . . . . . 7 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
54nfab 2952 . . . . . 6 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
63, 5nfcxfr 2945 . . . . 5 𝑑𝐶
76nfcri 2941 . . . 4 𝑑 𝑓𝐶
8 nfv 2008 . . . 4 𝑑 𝑥 ∈ dom 𝑓
92, 7, 8nf3an 1996 . . 3 𝑑(𝜑𝑓𝐶𝑥 ∈ dom 𝑓)
101, 9nfxfr 1941 . 2 𝑑𝜓
1110nf5ri 2232 1 (𝜓 → ∀𝑑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100  wal 1635   = wceq 1637  wcel 2158  {cab 2791  wral 3095  wrex 3096  wss 3766  cop 4373   cuni 4626  dom cdm 5308  cres 5310   Fn wfn 6093  cfv 6098   predc-bnj14 31076   FrSe w-bnj15 31080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-rex 3101
This theorem is referenced by:  bnj1501  31455
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