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Theorem bnj1321 35042
Description: Technical lemma for bnj60 35077. 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
bnj1321.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1321.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1321.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1321.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
Assertion
Ref Expression
bnj1321 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓   𝑅,𝑑,𝑓,𝑥
Allowed substitution hints:   𝜏(𝑥,𝑓,𝑑)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐺(𝑥)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj1321
Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 484 . 2 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃𝑓𝜏)
2 simp1 1136 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑅 FrSe 𝐴)
3 bnj1321.4 . . . . . . . . 9 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
43simplbi 497 . . . . . . . 8 (𝜏𝑓𝐶)
543ad2ant2 1134 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓𝐶)
6 bnj1321.3 . . . . . . . . . . . . 13 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
7 nfab1 2906 . . . . . . . . . . . . 13 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
86, 7nfcxfr 2902 . . . . . . . . . . . 12 𝑓𝐶
98nfcri 2896 . . . . . . . . . . 11 𝑓 𝑔𝐶
10 nfv 1913 . . . . . . . . . . 11 𝑓dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10nfan 1898 . . . . . . . . . 10 𝑓(𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12 eleq1w 2823 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
13 dmeq 5913 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
1413eqeq1d 2738 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
1512, 14anbi12d 632 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
163, 15bitrid 283 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝜏 ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
1711, 16sbiev 2313 . . . . . . . . 9 ([𝑔 / 𝑓]𝜏 ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
1817simplbi 497 . . . . . . . 8 ([𝑔 / 𝑓]𝜏𝑔𝐶)
19183ad2ant3 1135 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑔𝐶)
20 bnj1321.1 . . . . . . . 8 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
21 bnj1321.2 . . . . . . . 8 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
22 eqid 2736 . . . . . . . 8 (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
2320, 21, 6, 22bnj1326 35041 . . . . . . 7 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
242, 5, 19, 23syl3anc 1372 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
253simprbi 496 . . . . . . . . . 10 (𝜏 → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
26253ad2ant2 1134 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2717simprbi 496 . . . . . . . . . 10 ([𝑔 / 𝑓]𝜏 → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
28273ad2ant3 1135 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2926, 28eqtr4d 2779 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = dom 𝑔)
30 bnj1322 34837 . . . . . . . . 9 (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑓)
3130reseq2d 5996 . . . . . . . 8 (dom 𝑓 = dom 𝑔 → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑓 ↾ dom 𝑓))
3229, 31syl 17 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑓 ↾ dom 𝑓))
33 releq 5785 . . . . . . . . 9 (𝑧 = 𝑓 → (Rel 𝑧 ↔ Rel 𝑓))
3420, 21, 6bnj66 34875 . . . . . . . . 9 (𝑧𝐶 → Rel 𝑧)
3533, 34vtoclga 3576 . . . . . . . 8 (𝑓𝐶 → Rel 𝑓)
36 resdm 6043 . . . . . . . 8 (Rel 𝑓 → (𝑓 ↾ dom 𝑓) = 𝑓)
375, 35, 363syl 18 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ dom 𝑓) = 𝑓)
3832, 37eqtrd 2776 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑓)
39 eqeq2 2748 . . . . . . . . . 10 (dom 𝑓 = dom 𝑔 → ((dom 𝑓 ∩ dom 𝑔) = dom 𝑓 ↔ (dom 𝑓 ∩ dom 𝑔) = dom 𝑔))
4030, 39mpbid 232 . . . . . . . . 9 (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑔)
4140reseq2d 5996 . . . . . . . 8 (dom 𝑓 = dom 𝑔 → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ dom 𝑔))
4229, 41syl 17 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ dom 𝑔))
4320, 21, 6bnj66 34875 . . . . . . . 8 (𝑔𝐶 → Rel 𝑔)
44 resdm 6043 . . . . . . . 8 (Rel 𝑔 → (𝑔 ↾ dom 𝑔) = 𝑔)
4519, 43, 443syl 18 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ dom 𝑔) = 𝑔)
4642, 45eqtrd 2776 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑔)
4724, 38, 463eqtr3d 2784 . . . . 5 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)
48473expib 1122 . . . 4 (𝑅 FrSe 𝐴 → ((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
4948alrimivv 1927 . . 3 (𝑅 FrSe 𝐴 → ∀𝑓𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
5049adantr 480 . 2 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∀𝑓𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
51 nfv 1913 . . 3 𝑔𝜏
5251eu2 2608 . 2 (∃!𝑓𝜏 ↔ (∃𝑓𝜏 ∧ ∀𝑓𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)))
531, 50, 52sylanbrc 583 1 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1537   = wceq 1539  wex 1778  [wsb 2063  wcel 2107  ∃!weu 2567  {cab 2713  wral 3060  wrex 3069  cun 3948  cin 3949  wss 3950  {csn 4625  cop 4631  dom cdm 5684  cres 5686  Rel wrel 5689   Fn wfn 6555  cfv 6560   predc-bnj14 34703   FrSe w-bnj15 34707   trClc-bnj18 34709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-1o 8507  df-bnj17 34702  df-bnj14 34704  df-bnj13 34706  df-bnj15 34708  df-bnj18 34710  df-bnj19 34712
This theorem is referenced by:  bnj1489  35071
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