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Theorem bnj1312 32609
Description: Technical lemma for bnj60 32613. 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 501 . . . . . . 7 (𝜓𝑅 FrSe 𝐴)
41ssrab3 3971 . . . . . . . 8 𝐷𝐴
54a1i 11 . . . . . . 7 (𝜓𝐷𝐴)
62simprbi 500 . . . . . . 7 (𝜓𝐷 ≠ ∅)
71bnj1230 32353 . . . . . . . 8 (𝑤𝐷 → ∀𝑥 𝑤𝐷)
87bnj1228 32562 . . . . . . 7 ((𝑅 FrSe 𝐴𝐷𝐴𝐷 ≠ ∅) → ∃𝑥𝐷𝑦𝐷 ¬ 𝑦𝑅𝑥)
93, 5, 6, 8syl3anc 1372 . . . . . 6 (𝜓 → ∃𝑥𝐷𝑦𝐷 ¬ 𝑦𝑅𝑥)
10 bnj1312.7 . . . . . 6 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
11 nfv 1921 . . . . . . . . 9 𝑥 𝑅 FrSe 𝐴
127nfcii 2883 . . . . . . . . . 10 𝑥𝐷
13 nfcv 2899 . . . . . . . . . 10 𝑥
1412, 13nfne 3034 . . . . . . . . 9 𝑥 𝐷 ≠ ∅
1511, 14nfan 1906 . . . . . . . 8 𝑥(𝑅 FrSe 𝐴𝐷 ≠ ∅)
162, 15nfxfr 1859 . . . . . . 7 𝑥𝜓
1716nf5ri 2197 . . . . . 6 (𝜓 → ∀𝑥𝜓)
189, 10, 17bnj1521 32402 . . . . 5 (𝜓 → ∃𝑥𝜒)
1910simp2bi 1147 . . . . 5 (𝜒𝑥𝐷)
201bnj1538 32406 . . . . . 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 32607 . . . . . . . 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 32309 . . . . . . . . . . . . . 14 (𝜒𝑅 FrSe 𝐴)
3421, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1384 32583 . . . . . . . . . . . . . 14 (𝑅 FrSe 𝐴 → Fun 𝑃)
3533, 34syl 17 . . . . . . . . . . . . 13 (𝜒 → Fun 𝑃)
3621, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1415 32589 . . . . . . . . . . . . 13 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
3735, 36bnj1422 32388 . . . . . . . . . . . 12 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
3821, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 36bnj1416 32590 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
3921, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 35, 38, 36bnj1421 32593 . . . . . . . . . . . . 13 (𝜒 → Fun 𝑄)
4039, 38bnj1422 32388 . . . . . . . . . . . 12 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4121, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 37, 40bnj1423 32602 . . . . . . . . . . 11 (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
4232fneq2i 6436 . . . . . . . . . . . 12 (𝑄 Fn 𝐸𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4340, 42sylibr 237 . . . . . . . . . . 11 (𝜒𝑄 Fn 𝐸)
4421, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32bnj1452 32603 . . . . . . . . . . 11 (𝜒𝐸𝐵)
4521, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 30, 41, 43, 44bnj1463 32606 . . . . . . . . . 10 (𝜒𝑄𝐶)
4645, 38jca 515 . . . . . . . . 9 (𝜒 → (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4721, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 46bnj1491 32608 . . . . . . . 8 ((𝜒𝑄 ∈ V) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4830, 47mpdan 687 . . . . . . 7 (𝜒 → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4948, 24bnj1198 32346 . . . . . 6 (𝜒 → ∃𝑓𝜏)
5020, 49nsyl3 140 . . . . 5 (𝜒 → ¬ 𝑥𝐷)
5118, 19, 50bnj1304 32370 . . . 4 ¬ 𝜓
522, 51bnj1541 32407 . . 3 (𝑅 FrSe 𝐴𝐷 = ∅)
531, 52bnj1476 32398 . 2 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝜏)
5424exbii 1854 . . . 4 (∃𝑓𝜏 ↔ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
55 df-rex 3059 . . . 4 (∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5654, 55bitr4i 281 . . 3 (∃𝑓𝜏 ↔ ∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
5756ralbii 3080 . 2 (∀𝑥𝐴𝑓𝜏 ↔ ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
5853, 57sylib 221 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 209  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  {cab 2716  wne 2934  wral 3053  wrex 3054  {crab 3057  Vcvv 3398  [wsbc 3680  cun 3841  wss 3843  c0 4211  {csn 4516  cop 4522   cuni 4796   class class class wbr 5030  dom cdm 5525  cres 5527  Fun wfun 6333   Fn wfn 6334  cfv 6339   predc-bnj14 32237   FrSe w-bnj15 32241   trClc-bnj18 32243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-reg 9129  ax-inf2 9177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-1o 8131  df-bnj17 32236  df-bnj14 32238  df-bnj13 32240  df-bnj15 32242  df-bnj18 32244  df-bnj19 32246
This theorem is referenced by:  bnj1493  32610
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