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Theorem bnj1312 30826
Description: Technical lemma for bnj60 30830. 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
bnj1312.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1312.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1312.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1312.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1312.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1312.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1312.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1312.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1312.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1312.10 𝑃 = 𝐻
bnj1312.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1312.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1312.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1312.14 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
Assertion
Ref Expression
bnj1312 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥,𝑦,𝑧   𝐵,𝑓   𝑦,𝐶   𝑦,𝐷   𝐸,𝑑,𝑓,𝑦,𝑧   𝐺,𝑑,𝑓,𝑥,𝑦,𝑧   𝑧,𝑄   𝑅,𝑑,𝑓,𝑥,𝑦,𝑧   𝑧,𝑌   𝜒,𝑧   𝜓,𝑦   𝜏,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑓,𝑑)   𝐸(𝑥)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1312
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1312.5 . . 3 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2 bnj1312.6 . . . 4 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
32simplbi 476 . . . . . . 7 (𝜓𝑅 FrSe 𝐴)
41bnj21 30483 . . . . . . . 8 𝐷𝐴
54a1i 11 . . . . . . 7 (𝜓𝐷𝐴)
62simprbi 480 . . . . . . 7 (𝜓𝐷 ≠ ∅)
71bnj1230 30573 . . . . . . . 8 (𝑤𝐷 → ∀𝑥 𝑤𝐷)
87bnj1228 30779 . . . . . . 7 ((𝑅 FrSe 𝐴𝐷𝐴𝐷 ≠ ∅) → ∃𝑥𝐷𝑦𝐷 ¬ 𝑦𝑅𝑥)
93, 5, 6, 8syl3anc 1323 . . . . . 6 (𝜓 → ∃𝑥𝐷𝑦𝐷 ¬ 𝑦𝑅𝑥)
10 bnj1312.7 . . . . . 6 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
11 nfv 1845 . . . . . . . . 9 𝑥 𝑅 FrSe 𝐴
127nfcii 2758 . . . . . . . . . 10 𝑥𝐷
13 nfcv 2767 . . . . . . . . . 10 𝑥
1412, 13nfne 2896 . . . . . . . . 9 𝑥 𝐷 ≠ ∅
1511, 14nfan 1830 . . . . . . . 8 𝑥(𝑅 FrSe 𝐴𝐷 ≠ ∅)
162, 15nfxfr 1777 . . . . . . 7 𝑥𝜓
1716nf5ri 2068 . . . . . 6 (𝜓 → ∀𝑥𝜓)
189, 10, 17bnj1521 30621 . . . . 5 (𝜓 → ∃𝑥𝜒)
1910simp2bi 1075 . . . . 5 (𝜒𝑥𝐷)
201bnj1538 30625 . . . . . 6 (𝑥𝐷 → ¬ ∃𝑓𝜏)
21 bnj1312.1 . . . . . . . . 9 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
22 bnj1312.2 . . . . . . . . 9 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
23 bnj1312.3 . . . . . . . . 9 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
24 bnj1312.4 . . . . . . . . 9 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
25 bnj1312.8 . . . . . . . . 9 (𝜏′[𝑦 / 𝑥]𝜏)
26 bnj1312.9 . . . . . . . . 9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
27 bnj1312.10 . . . . . . . . 9 𝑃 = 𝐻
28 bnj1312.11 . . . . . . . . 9 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
29 bnj1312.12 . . . . . . . . 9 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3021, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29bnj1489 30824 . . . . . . . 8 (𝜒𝑄 ∈ V)
31 bnj1312.13 . . . . . . . . . . 11 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
32 bnj1312.14 . . . . . . . . . . 11 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
3310, 3bnj835 30529 . . . . . . . . . . . . . 14 (𝜒𝑅 FrSe 𝐴)
3421, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1384 30800 . . . . . . . . . . . . . 14 (𝑅 FrSe 𝐴 → Fun 𝑃)
3533, 34syl 17 . . . . . . . . . . . . 13 (𝜒 → Fun 𝑃)
3621, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1415 30806 . . . . . . . . . . . . 13 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
3735, 36bnj1422 30608 . . . . . . . . . . . 12 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
3821, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 36bnj1416 30807 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
3921, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 35, 38, 36bnj1421 30810 . . . . . . . . . . . . 13 (𝜒 → Fun 𝑄)
4039, 38bnj1422 30608 . . . . . . . . . . . 12 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4121, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 37, 40bnj1423 30819 . . . . . . . . . . 11 (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
4232fneq2i 5946 . . . . . . . . . . . 12 (𝑄 Fn 𝐸𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4340, 42sylibr 224 . . . . . . . . . . 11 (𝜒𝑄 Fn 𝐸)
4421, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32bnj1452 30820 . . . . . . . . . . 11 (𝜒𝐸𝐵)
4521, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 30, 41, 43, 44bnj1463 30823 . . . . . . . . . 10 (𝜒𝑄𝐶)
4645, 38jca 554 . . . . . . . . 9 (𝜒 → (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4721, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 46bnj1491 30825 . . . . . . . 8 ((𝜒𝑄 ∈ V) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4830, 47mpdan 701 . . . . . . 7 (𝜒 → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4948, 24bnj1198 30566 . . . . . 6 (𝜒 → ∃𝑓𝜏)
5020, 49nsyl3 133 . . . . 5 (𝜒 → ¬ 𝑥𝐷)
5118, 19, 50bnj1304 30590 . . . 4 ¬ 𝜓
522, 51bnj1541 30626 . . 3 (𝑅 FrSe 𝐴𝐷 = ∅)
531, 52bnj1476 30617 . 2 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝜏)
5424exbii 1772 . . . 4 (∃𝑓𝜏 ↔ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
55 df-rex 2918 . . . 4 (∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5654, 55bitr4i 267 . . 3 (∃𝑓𝜏 ↔ ∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
5756ralbii 2979 . 2 (∀𝑥𝐴𝑓𝜏 ↔ ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
5853, 57sylib 208 1 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 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 1992  {cab 2612  wne 2796  wral 2912  wrex 2913  {crab 2916  Vcvv 3191  [wsbc 3422  cun 3558  wss 3560  c0 3896  {csn 4153  cop 4159   cuni 4407   class class class wbr 4618  dom cdm 5079  cres 5081  Fun wfun 5844   Fn wfn 5845  cfv 5850   predc-bnj14 30453   FrSe w-bnj15 30457   trClc-bnj18 30459
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-reg 8442  ax-inf2 8483
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-1o 7506  df-bnj17 30452  df-bnj14 30454  df-bnj13 30456  df-bnj15 30458  df-bnj18 30460  df-bnj19 30462
This theorem is referenced by:  bnj1493  30827
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