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Theorem bnj1312 31595
Description: Technical lemma for bnj60 31599. 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 491 . . . . . . 7 (𝜓𝑅 FrSe 𝐴)
41ssrab3 3850 . . . . . . . 8 𝐷𝐴
54a1i 11 . . . . . . 7 (𝜓𝐷𝐴)
62simprbi 490 . . . . . . 7 (𝜓𝐷 ≠ ∅)
71bnj1230 31342 . . . . . . . 8 (𝑤𝐷 → ∀𝑥 𝑤𝐷)
87bnj1228 31548 . . . . . . 7 ((𝑅 FrSe 𝐴𝐷𝐴𝐷 ≠ ∅) → ∃𝑥𝐷𝑦𝐷 ¬ 𝑦𝑅𝑥)
93, 5, 6, 8syl3anc 1490 . . . . . 6 (𝜓 → ∃𝑥𝐷𝑦𝐷 ¬ 𝑦𝑅𝑥)
10 bnj1312.7 . . . . . 6 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
11 nfv 2009 . . . . . . . . 9 𝑥 𝑅 FrSe 𝐴
127nfcii 2898 . . . . . . . . . 10 𝑥𝐷
13 nfcv 2907 . . . . . . . . . 10 𝑥
1412, 13nfne 3037 . . . . . . . . 9 𝑥 𝐷 ≠ ∅
1511, 14nfan 1998 . . . . . . . 8 𝑥(𝑅 FrSe 𝐴𝐷 ≠ ∅)
162, 15nfxfr 1948 . . . . . . 7 𝑥𝜓
1716nf5ri 2227 . . . . . 6 (𝜓 → ∀𝑥𝜓)
189, 10, 17bnj1521 31390 . . . . 5 (𝜓 → ∃𝑥𝜒)
1910simp2bi 1176 . . . . 5 (𝜒𝑥𝐷)
201bnj1538 31394 . . . . . 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 31593 . . . . . . . 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 31298 . . . . . . . . . . . . . 14 (𝜒𝑅 FrSe 𝐴)
3421, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1384 31569 . . . . . . . . . . . . . 14 (𝑅 FrSe 𝐴 → Fun 𝑃)
3533, 34syl 17 . . . . . . . . . . . . 13 (𝜒 → Fun 𝑃)
3621, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1415 31575 . . . . . . . . . . . . 13 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
3735, 36bnj1422 31377 . . . . . . . . . . . 12 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
3821, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 36bnj1416 31576 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
3921, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 35, 38, 36bnj1421 31579 . . . . . . . . . . . . 13 (𝜒 → Fun 𝑄)
4039, 38bnj1422 31377 . . . . . . . . . . . 12 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4121, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 37, 40bnj1423 31588 . . . . . . . . . . 11 (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
4232fneq2i 6166 . . . . . . . . . . . 12 (𝑄 Fn 𝐸𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4340, 42sylibr 225 . . . . . . . . . . 11 (𝜒𝑄 Fn 𝐸)
4421, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32bnj1452 31589 . . . . . . . . . . 11 (𝜒𝐸𝐵)
4521, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 30, 41, 43, 44bnj1463 31592 . . . . . . . . . 10 (𝜒𝑄𝐶)
4645, 38jca 507 . . . . . . . . 9 (𝜒 → (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4721, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 46bnj1491 31594 . . . . . . . 8 ((𝜒𝑄 ∈ V) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4830, 47mpdan 678 . . . . . . 7 (𝜒 → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4948, 24bnj1198 31335 . . . . . 6 (𝜒 → ∃𝑓𝜏)
5020, 49nsyl3 135 . . . . 5 (𝜒 → ¬ 𝑥𝐷)
5118, 19, 50bnj1304 31359 . . . 4 ¬ 𝜓
522, 51bnj1541 31395 . . 3 (𝑅 FrSe 𝐴𝐷 = ∅)
531, 52bnj1476 31386 . 2 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝜏)
5424exbii 1943 . . . 4 (∃𝑓𝜏 ↔ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
55 df-rex 3061 . . . 4 (∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5654, 55bitr4i 269 . . 3 (∃𝑓𝜏 ↔ ∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
5756ralbii 3127 . 2 (∀𝑥𝐴𝑓𝜏 ↔ ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
5853, 57sylib 209 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 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  [wsbc 3598  cun 3732  wss 3734  c0 4081  {csn 4336  cop 4342   cuni 4596   class class class wbr 4811  dom cdm 5279  cres 5281  Fun wfun 6064   Fn wfn 6065  cfv 6070   predc-bnj14 31226   FrSe w-bnj15 31230   trClc-bnj18 31232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-reg 8708  ax-inf2 8757
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-om 7268  df-1o 7768  df-bnj17 31225  df-bnj14 31227  df-bnj13 31229  df-bnj15 31231  df-bnj18 31233  df-bnj19 31235
This theorem is referenced by:  bnj1493  31596
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