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Theorem bnj1491 30830
Description: Technical lemma for bnj60 30835. 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 30826 . . . 4 (𝑤𝑄 → ∀𝑓 𝑤𝑄)
1514nfcii 2752 . . 3 𝑓𝑄
164bnj1317 30597 . . . . . 6 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1716nfcii 2752 . . . . 5 𝑓𝐶
1815, 17nfel 2773 . . . 4 𝑓 𝑄𝐶
1915nfdm 5327 . . . . 5 𝑓dom 𝑄
2019nfeq1 2774 . . . 4 𝑓dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
2118, 20nfan 1825 . . 3 𝑓(𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
22 eleq1 2686 . . . 4 (𝑓 = 𝑄 → (𝑓𝐶𝑄𝐶))
23 dmeq 5284 . . . . 5 (𝑓 = 𝑄 → dom 𝑓 = dom 𝑄)
2423eqeq1d 2623 . . . 4 (𝑓 = 𝑄 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
2522, 24anbi12d 746 . . 3 (𝑓 = 𝑄 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
2615, 21, 25spcegf 3275 . 2 (𝑄 ∈ V → ((𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
271, 26mpan9 486 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 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  [wsbc 3417  cun 3553  wss 3555  c0 3891  {csn 4148  cop 4154   cuni 4402   class class class wbr 4613  dom cdm 5074  cres 5076   Fn wfn 5842  cfv 5847   predc-bnj14 30458   FrSe w-bnj15 30462   trClc-bnj18 30464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-dm 5084  df-res 5086  df-iota 5810  df-fv 5855
This theorem is referenced by:  bnj1312  30831
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