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Theorem bnj1321 30824
Description: Technical lemma for bnj60 30859. 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 2763 . . . . . . . . . . . . 13 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
86, 7nfcxfr 2759 . . . . . . . . . . . 12 𝑓𝐶
98nfcri 2755 . . . . . . . . . . 11 𝑓 𝑔𝐶
10 nfv 1840 . . . . . . . . . . 11 𝑓dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10nfan 1825 . . . . . . . . . 10 𝑓(𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12 eleq1 2686 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
13 dmeq 5286 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
1413eqeq1d 2623 . . . . . . . . . . . 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 2407 . . . . . . . . 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 2621 . . . . . . . 8 (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
2320, 21, 6, 22bnj1326 30823 . . . . . . 7 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
242, 5, 19, 23syl3anc 1323 . . . . . 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 2658 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = dom 𝑔)
30 bnj1322 30622 . . . . . . . . 9 (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑓)
3130reseq2d 5358 . . . . . . . 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 5164 . . . . . . . . 9 (𝑧 = 𝑓 → (Rel 𝑧 ↔ Rel 𝑓))
3420, 21, 6bnj66 30659 . . . . . . . . 9 (𝑧𝐶 → Rel 𝑧)
3533, 34vtoclga 3258 . . . . . . . 8 (𝑓𝐶 → Rel 𝑓)
36 resdm 5402 . . . . . . . 8 (Rel 𝑓 → (𝑓 ↾ dom 𝑓) = 𝑓)
375, 35, 363syl 18 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ dom 𝑓) = 𝑓)
3832, 37eqtrd 2655 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑓)
39 eqeq2 2632 . . . . . . . . . 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 5358 . . . . . . . 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 30659 . . . . . . . 8 (𝑔𝐶 → Rel 𝑔)
44 resdm 5402 . . . . . . . 8 (Rel 𝑔 → (𝑔 ↾ dom 𝑔) = 𝑔)
4519, 43, 443syl 18 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ dom 𝑔) = 𝑔)
4642, 45eqtrd 2655 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑔)
4724, 38, 463eqtr3d 2663 . . . . 5 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)
48473expib 1265 . . . 4 (𝑅 FrSe 𝐴 → ((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
4948alrimivv 1853 . . 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 1840 . . 3 𝑔𝜏
5251eu2 2508 . 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 1478   = wceq 1480  wex 1701  [wsb 1877  wcel 1987  ∃!weu 2469  {cab 2607  wral 2907  wrex 2908  cun 3554  cin 3555  wss 3556  {csn 4150  cop 4156  dom cdm 5076  cres 5078  Rel wrel 5081   Fn wfn 5844  cfv 5849   predc-bnj14 30482   FrSe w-bnj15 30486   trClc-bnj18 30488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-reg 8444  ax-inf2 8485
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-om 7016  df-1o 7508  df-bnj17 30481  df-bnj14 30483  df-bnj13 30485  df-bnj15 30487  df-bnj18 30489  df-bnj19 30491
This theorem is referenced by:  bnj1489  30853
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