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Theorem bnj1491 32414
Description: Technical lemma for bnj60 32419. 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
bnj1491.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1491.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1491.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1491.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1491.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1491.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1491.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1491.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1491.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1491.10 𝑃 = 𝐻
bnj1491.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1491.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1491.13 (𝜒 → (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
Assertion
Ref Expression
bnj1491 ((𝜒𝑄 ∈ V) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑅,𝑓   𝑥,𝑓
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑓,𝑑)   𝐴(𝑥,𝑦,𝑑)   𝐵(𝑥,𝑦,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑥,𝑦,𝑑)   𝐺(𝑥,𝑦,𝑑)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1491
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1491.13 . 2 (𝜒 → (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
2 bnj1491.1 . . . . 5 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
3 bnj1491.2 . . . . 5 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4 bnj1491.3 . . . . 5 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
5 bnj1491.4 . . . . 5 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
6 bnj1491.5 . . . . 5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
7 bnj1491.6 . . . . 5 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
8 bnj1491.7 . . . . 5 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
9 bnj1491.8 . . . . 5 (𝜏′[𝑦 / 𝑥]𝜏)
10 bnj1491.9 . . . . 5 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
11 bnj1491.10 . . . . 5 𝑃 = 𝐻
12 bnj1491.11 . . . . 5 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
13 bnj1491.12 . . . . 5 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
142, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13bnj1466 32410 . . . 4 (𝑤𝑄 → ∀𝑓 𝑤𝑄)
1514nfcii 2966 . . 3 𝑓𝑄
164bnj1317 32178 . . . . . 6 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1716nfcii 2966 . . . . 5 𝑓𝐶
1815, 17nfel 2996 . . . 4 𝑓 𝑄𝐶
1915nfdm 5811 . . . . 5 𝑓dom 𝑄
2019nfeq1 2997 . . . 4 𝑓dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
2118, 20nfan 1901 . . 3 𝑓(𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
22 eleq1 2903 . . . 4 (𝑓 = 𝑄 → (𝑓𝐶𝑄𝐶))
23 dmeq 5760 . . . . 5 (𝑓 = 𝑄 → dom 𝑓 = dom 𝑄)
2423eqeq1d 2826 . . . 4 (𝑓 = 𝑄 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
2522, 24anbi12d 633 . . 3 (𝑓 = 𝑄 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
2615, 21, 25spcegf 3577 . 2 (𝑄 ∈ V → ((𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
271, 26mpan9 510 1 ((𝜒𝑄 ∈ V) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wne 3014  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  [wsbc 3758  cun 3917  wss 3919  c0 4276  {csn 4550  cop 4556   cuni 4824   class class class wbr 5053  dom cdm 5543  cres 5545   Fn wfn 6340  cfv 6345   predc-bnj14 32043   FrSe w-bnj15 32047   trClc-bnj18 32049
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-xp 5549  df-dm 5553  df-res 5555  df-iota 6304  df-fv 6353
This theorem is referenced by:  bnj1312  32415
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