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Theorem bnj1321 31069
Description: Technical lemma for bnj60 31104. 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 477 . 2 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃𝑓𝜏)
2 simp1 1059 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑅 FrSe 𝐴)
3 bnj1321.4 . . . . . . . . 9 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
43simplbi 476 . . . . . . . 8 (𝜏𝑓𝐶)
543ad2ant2 1081 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓𝐶)
6 bnj1321.3 . . . . . . . . . . . . 13 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
7 nfab1 2764 . . . . . . . . . . . . 13 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
86, 7nfcxfr 2760 . . . . . . . . . . . 12 𝑓𝐶
98nfcri 2756 . . . . . . . . . . 11 𝑓 𝑔𝐶
10 nfv 1841 . . . . . . . . . . 11 𝑓dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10nfan 1826 . . . . . . . . . 10 𝑓(𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12 eleq1 2687 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
13 dmeq 5313 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
1413eqeq1d 2622 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
1512, 14anbi12d 746 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
163, 15syl5bb 272 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝜏 ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
1711, 16sbie 2406 . . . . . . . . 9 ([𝑔 / 𝑓]𝜏 ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
1817simplbi 476 . . . . . . . 8 ([𝑔 / 𝑓]𝜏𝑔𝐶)
19183ad2ant3 1082 . . . . . . 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 2620 . . . . . . . 8 (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
2320, 21, 6, 22bnj1326 31068 . . . . . . 7 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
242, 5, 19, 23syl3anc 1324 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
253simprbi 480 . . . . . . . . . 10 (𝜏 → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
26253ad2ant2 1081 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2717simprbi 480 . . . . . . . . . 10 ([𝑔 / 𝑓]𝜏 → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
28273ad2ant3 1082 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2926, 28eqtr4d 2657 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = dom 𝑔)
30 bnj1322 30867 . . . . . . . . 9 (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑓)
3130reseq2d 5385 . . . . . . . 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 5191 . . . . . . . . 9 (𝑧 = 𝑓 → (Rel 𝑧 ↔ Rel 𝑓))
3420, 21, 6bnj66 30904 . . . . . . . . 9 (𝑧𝐶 → Rel 𝑧)
3533, 34vtoclga 3267 . . . . . . . 8 (𝑓𝐶 → Rel 𝑓)
36 resdm 5429 . . . . . . . 8 (Rel 𝑓 → (𝑓 ↾ dom 𝑓) = 𝑓)
375, 35, 363syl 18 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ dom 𝑓) = 𝑓)
3832, 37eqtrd 2654 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑓)
39 eqeq2 2631 . . . . . . . . . 10 (dom 𝑓 = dom 𝑔 → ((dom 𝑓 ∩ dom 𝑔) = dom 𝑓 ↔ (dom 𝑓 ∩ dom 𝑔) = dom 𝑔))
4030, 39mpbid 222 . . . . . . . . 9 (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑔)
4140reseq2d 5385 . . . . . . . 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 30904 . . . . . . . 8 (𝑔𝐶 → Rel 𝑔)
44 resdm 5429 . . . . . . . 8 (Rel 𝑔 → (𝑔 ↾ dom 𝑔) = 𝑔)
4519, 43, 443syl 18 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ dom 𝑔) = 𝑔)
4642, 45eqtrd 2654 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑔)
4724, 38, 463eqtr3d 2662 . . . . 5 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)
48473expib 1266 . . . 4 (𝑅 FrSe 𝐴 → ((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
4948alrimivv 1854 . . 3 (𝑅 FrSe 𝐴 → ∀𝑓𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
5049adantr 481 . 2 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∀𝑓𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
51 nfv 1841 . . 3 𝑔𝜏
5251eu2 2507 . 2 (∃!𝑓𝜏 ↔ (∃𝑓𝜏 ∧ ∀𝑓𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)))
531, 50, 52sylanbrc 697 1 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1479   = wceq 1481  wex 1702  [wsb 1878  wcel 1988  ∃!weu 2468  {cab 2606  wral 2909  wrex 2910  cun 3565  cin 3566  wss 3567  {csn 4168  cop 4174  dom cdm 5104  cres 5106  Rel wrel 5109   Fn wfn 5871  cfv 5876   predc-bnj14 30728   FrSe w-bnj15 30732   trClc-bnj18 30734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-reg 8482  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-1o 7545  df-bnj17 30727  df-bnj14 30729  df-bnj13 30731  df-bnj15 30733  df-bnj18 30735  df-bnj19 30737
This theorem is referenced by:  bnj1489  31098
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