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Theorem bnj1312 33670
Description: Technical lemma for bnj60 33674. 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 498 . . . . . . 7 (𝜓𝑅 FrSe 𝐴)
41ssrab3 4040 . . . . . . . 8 𝐷𝐴
54a1i 11 . . . . . . 7 (𝜓𝐷𝐴)
62simprbi 497 . . . . . . 7 (𝜓𝐷 ≠ ∅)
71bnj1230 33414 . . . . . . . 8 (𝑤𝐷 → ∀𝑥 𝑤𝐷)
87bnj1228 33623 . . . . . . 7 ((𝑅 FrSe 𝐴𝐷𝐴𝐷 ≠ ∅) → ∃𝑥𝐷𝑦𝐷 ¬ 𝑦𝑅𝑥)
93, 5, 6, 8syl3anc 1371 . . . . . 6 (𝜓 → ∃𝑥𝐷𝑦𝐷 ¬ 𝑦𝑅𝑥)
10 bnj1312.7 . . . . . 6 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
11 nfv 1917 . . . . . . . . 9 𝑥 𝑅 FrSe 𝐴
127nfcii 2891 . . . . . . . . . 10 𝑥𝐷
13 nfcv 2907 . . . . . . . . . 10 𝑥
1412, 13nfne 3045 . . . . . . . . 9 𝑥 𝐷 ≠ ∅
1511, 14nfan 1902 . . . . . . . 8 𝑥(𝑅 FrSe 𝐴𝐷 ≠ ∅)
162, 15nfxfr 1855 . . . . . . 7 𝑥𝜓
1716nf5ri 2188 . . . . . 6 (𝜓 → ∀𝑥𝜓)
189, 10, 17bnj1521 33463 . . . . 5 (𝜓 → ∃𝑥𝜒)
1910simp2bi 1146 . . . . 5 (𝜒𝑥𝐷)
201bnj1538 33467 . . . . . 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 33668 . . . . . . . 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 33371 . . . . . . . . . . . . . 14 (𝜒𝑅 FrSe 𝐴)
3421, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1384 33644 . . . . . . . . . . . . . 14 (𝑅 FrSe 𝐴 → Fun 𝑃)
3533, 34syl 17 . . . . . . . . . . . . 13 (𝜒 → Fun 𝑃)
3621, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1415 33650 . . . . . . . . . . . . 13 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
3735, 36bnj1422 33449 . . . . . . . . . . . 12 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
3821, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 36bnj1416 33651 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
3921, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 35, 38, 36bnj1421 33654 . . . . . . . . . . . . 13 (𝜒 → Fun 𝑄)
4039, 38bnj1422 33449 . . . . . . . . . . . 12 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4121, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 37, 40bnj1423 33663 . . . . . . . . . . 11 (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
4232fneq2i 6600 . . . . . . . . . . . 12 (𝑄 Fn 𝐸𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4340, 42sylibr 233 . . . . . . . . . . 11 (𝜒𝑄 Fn 𝐸)
4421, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32bnj1452 33664 . . . . . . . . . . 11 (𝜒𝐸𝐵)
4521, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 30, 41, 43, 44bnj1463 33667 . . . . . . . . . 10 (𝜒𝑄𝐶)
4645, 38jca 512 . . . . . . . . 9 (𝜒 → (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4721, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 46bnj1491 33669 . . . . . . . 8 ((𝜒𝑄 ∈ V) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4830, 47mpdan 685 . . . . . . 7 (𝜒 → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4948, 24bnj1198 33407 . . . . . 6 (𝜒 → ∃𝑓𝜏)
5020, 49nsyl3 138 . . . . 5 (𝜒 → ¬ 𝑥𝐷)
5118, 19, 50bnj1304 33431 . . . 4 ¬ 𝜓
522, 51bnj1541 33468 . . 3 (𝑅 FrSe 𝐴𝐷 = ∅)
531, 52bnj1476 33459 . 2 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝜏)
5424exbii 1850 . . . 4 (∃𝑓𝜏 ↔ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
55 df-rex 3074 . . . 4 (∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5654, 55bitr4i 277 . . 3 (∃𝑓𝜏 ↔ ∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
5756ralbii 3096 . 2 (∀𝑥𝐴𝑓𝜏 ↔ ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
5853, 57sylib 217 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  [wsbc 3739  cun 3908  wss 3910  c0 4282  {csn 4586  cop 4592   cuni 4865   class class class wbr 5105  dom cdm 5633  cres 5635  Fun wfun 6490   Fn wfn 6491  cfv 6496   predc-bnj14 33300   FrSe w-bnj15 33304   trClc-bnj18 33306
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-bnj17 33299  df-bnj14 33301  df-bnj13 33303  df-bnj15 33305  df-bnj18 33307  df-bnj19 33309
This theorem is referenced by:  bnj1493  33671
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