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Theorem bnj1489 30829
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
bnj1489.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1489.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1489.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1489.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1489.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1489.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1489.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1489.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1489.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1489.10 𝑃 = 𝐻
bnj1489.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1489.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
Assertion
Ref Expression
bnj1489 (𝜒𝑄 ∈ V)
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝑦,𝐴,𝑓,𝑥   𝐵,𝑓   𝑦,𝐷   𝐺,𝑑,𝑓   𝑅,𝑑,𝑓,𝑥   𝑦,𝑅   𝜓,𝑦   𝜏,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑑)   𝐷(𝑥,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑓,𝑑)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1489
StepHypRef Expression
1 bnj1489.12 . 2 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
2 bnj1489.10 . . . 4 𝑃 = 𝐻
3 bnj1489.7 . . . . . . . 8 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
4 bnj1489.6 . . . . . . . . 9 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
5 bnj1364 30801 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝑅 Se 𝐴)
6 df-bnj13 30461 . . . . . . . . . 10 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
75, 6sylib 208 . . . . . . . . 9 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
84, 7bnj832 30533 . . . . . . . 8 (𝜓 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
93, 8bnj835 30534 . . . . . . 7 (𝜒 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
10 bnj1489.5 . . . . . . . 8 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
1110, 3bnj1212 30575 . . . . . . 7 (𝜒𝑥𝐴)
129, 11bnj1294 30593 . . . . . 6 (𝜒 → pred(𝑥, 𝐴, 𝑅) ∈ V)
13 nfv 1840 . . . . . . . . 9 𝑦𝜓
14 nfv 1840 . . . . . . . . 9 𝑦 𝑥𝐷
15 nfra1 2936 . . . . . . . . 9 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
1613, 14, 15nf3an 1828 . . . . . . . 8 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
173, 16nfxfr 1776 . . . . . . 7 𝑦𝜒
184simplbi 476 . . . . . . . . . . 11 (𝜓𝑅 FrSe 𝐴)
193, 18bnj835 30534 . . . . . . . . . 10 (𝜒𝑅 FrSe 𝐴)
2019adantr 481 . . . . . . . . 9 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
21 bnj1489.1 . . . . . . . . . . 11 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
22 bnj1489.2 . . . . . . . . . . 11 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
23 bnj1489.3 . . . . . . . . . . 11 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
24 bnj1489.4 . . . . . . . . . . 11 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
25 bnj1489.8 . . . . . . . . . . 11 (𝜏′[𝑦 / 𝑥]𝜏)
2621, 22, 23, 24, 10, 4, 3, 25bnj1388 30806 . . . . . . . . . 10 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
2726r19.21bi 2927 . . . . . . . . 9 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
28 nfv 1840 . . . . . . . . . . . 12 𝑥 𝑅 FrSe 𝐴
29 nfsbc1v 3437 . . . . . . . . . . . . . 14 𝑥[𝑦 / 𝑥]𝜏
3025, 29nfxfr 1776 . . . . . . . . . . . . 13 𝑥𝜏′
3130nfex 2151 . . . . . . . . . . . 12 𝑥𝑓𝜏′
3228, 31nfan 1825 . . . . . . . . . . 11 𝑥(𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)
3330nfeu 2485 . . . . . . . . . . 11 𝑥∃!𝑓𝜏′
3432, 33nfim 1822 . . . . . . . . . 10 𝑥((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
35 sneq 4158 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → {𝑥} = {𝑦})
36 bnj1318 30798 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
3735, 36uneq12d 3746 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
3837eqeq2d 2631 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3938anbi2d 739 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
4021, 22, 23, 24, 25bnj1373 30803 . . . . . . . . . . . . . 14 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
4139, 40syl6bbr 278 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ 𝜏′))
4241exbidv 1847 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃𝑓𝜏′))
4342anbi2d 739 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) ↔ (𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)))
4441eubidv 2489 . . . . . . . . . . 11 (𝑥 = 𝑦 → (∃!𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃!𝑓𝜏′))
4543, 44imbi12d 334 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) → ∃!𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)))
46 biid 251 . . . . . . . . . . 11 ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4721, 22, 23, 46bnj1321 30800 . . . . . . . . . 10 ((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) → ∃!𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4834, 45, 47chvar 2261 . . . . . . . . 9 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
4920, 27, 48syl2anc 692 . . . . . . . 8 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃!𝑓𝜏′)
5049ex 450 . . . . . . 7 (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃!𝑓𝜏′))
5117, 50ralrimi 2951 . . . . . 6 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′)
52 bnj1489.9 . . . . . . 7 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
5352a1i 11 . . . . . 6 (𝜒𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′})
54 biid 251 . . . . . . 7 (( pred(𝑥, 𝐴, 𝑅) ∈ V ∧ ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}) ↔ ( pred(𝑥, 𝐴, 𝑅) ∈ V ∧ ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}))
5554bnj1366 30605 . . . . . 6 (( pred(𝑥, 𝐴, 𝑅) ∈ V ∧ ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}) → 𝐻 ∈ V)
5612, 51, 53, 55syl3anc 1323 . . . . 5 (𝜒𝐻 ∈ V)
57 uniexg 6908 . . . . 5 (𝐻 ∈ V → 𝐻 ∈ V)
5856, 57syl 17 . . . 4 (𝜒 𝐻 ∈ V)
592, 58syl5eqel 2702 . . 3 (𝜒𝑃 ∈ V)
60 snex 4869 . . . 4 {⟨𝑥, (𝐺𝑍)⟩} ∈ V
6160a1i 11 . . 3 (𝜒 → {⟨𝑥, (𝐺𝑍)⟩} ∈ V)
6259, 61bnj1149 30568 . 2 (𝜒 → (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}) ∈ V)
631, 62syl5eqel 2702 1 (𝜒𝑄 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  {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   Se w-bnj13 30460   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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-bnj17 30457  df-bnj14 30459  df-bnj13 30461  df-bnj15 30463  df-bnj18 30465  df-bnj19 30467
This theorem is referenced by:  bnj1312  30831
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