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Theorem bnj1321 34336
Description: Technical lemma for bnj60 34371. 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 483 . 2 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃𝑓𝜏)
2 simp1 1134 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑅 FrSe 𝐴)
3 bnj1321.4 . . . . . . . . 9 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
43simplbi 496 . . . . . . . 8 (𝜏𝑓𝐶)
543ad2ant2 1132 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓𝐶)
6 bnj1321.3 . . . . . . . . . . . . 13 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
7 nfab1 2903 . . . . . . . . . . . . 13 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
86, 7nfcxfr 2899 . . . . . . . . . . . 12 𝑓𝐶
98nfcri 2888 . . . . . . . . . . 11 𝑓 𝑔𝐶
10 nfv 1915 . . . . . . . . . . 11 𝑓dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10nfan 1900 . . . . . . . . . 10 𝑓(𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12 eleq1w 2814 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
13 dmeq 5902 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
1413eqeq1d 2732 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
1512, 14anbi12d 629 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
163, 15bitrid 282 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝜏 ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
1711, 16sbiev 2306 . . . . . . . . 9 ([𝑔 / 𝑓]𝜏 ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
1817simplbi 496 . . . . . . . 8 ([𝑔 / 𝑓]𝜏𝑔𝐶)
19183ad2ant3 1133 . . . . . . 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 2730 . . . . . . . 8 (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
2320, 21, 6, 22bnj1326 34335 . . . . . . 7 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
242, 5, 19, 23syl3anc 1369 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
253simprbi 495 . . . . . . . . . 10 (𝜏 → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
26253ad2ant2 1132 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2717simprbi 495 . . . . . . . . . 10 ([𝑔 / 𝑓]𝜏 → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
28273ad2ant3 1133 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2926, 28eqtr4d 2773 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → dom 𝑓 = dom 𝑔)
30 bnj1322 34131 . . . . . . . . 9 (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑓)
3130reseq2d 5980 . . . . . . . 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 5775 . . . . . . . . 9 (𝑧 = 𝑓 → (Rel 𝑧 ↔ Rel 𝑓))
3420, 21, 6bnj66 34169 . . . . . . . . 9 (𝑧𝐶 → Rel 𝑧)
3533, 34vtoclga 3565 . . . . . . . 8 (𝑓𝐶 → Rel 𝑓)
36 resdm 6025 . . . . . . . 8 (Rel 𝑓 → (𝑓 ↾ dom 𝑓) = 𝑓)
375, 35, 363syl 18 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ dom 𝑓) = 𝑓)
3832, 37eqtrd 2770 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑓)
39 eqeq2 2742 . . . . . . . . . 10 (dom 𝑓 = dom 𝑔 → ((dom 𝑓 ∩ dom 𝑔) = dom 𝑓 ↔ (dom 𝑓 ∩ dom 𝑔) = dom 𝑔))
4030, 39mpbid 231 . . . . . . . . 9 (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑔)
4140reseq2d 5980 . . . . . . . 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 34169 . . . . . . . 8 (𝑔𝐶 → Rel 𝑔)
44 resdm 6025 . . . . . . . 8 (Rel 𝑔 → (𝑔 ↾ dom 𝑔) = 𝑔)
4519, 43, 443syl 18 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ dom 𝑔) = 𝑔)
4642, 45eqtrd 2770 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)) = 𝑔)
4724, 38, 463eqtr3d 2778 . . . . 5 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)
48473expib 1120 . . . 4 (𝑅 FrSe 𝐴 → ((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
4948alrimivv 1929 . . 3 (𝑅 FrSe 𝐴 → ∀𝑓𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
5049adantr 479 . 2 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∀𝑓𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔))
51 nfv 1915 . . 3 𝑔𝜏
5251eu2 2603 . 2 (∃!𝑓𝜏 ↔ (∃𝑓𝜏 ∧ ∀𝑓𝑔((𝜏 ∧ [𝑔 / 𝑓]𝜏) → 𝑓 = 𝑔)))
531, 50, 52sylanbrc 581 1 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1085  wal 1537   = wceq 1539  wex 1779  [wsb 2065  wcel 2104  ∃!weu 2560  {cab 2707  wral 3059  wrex 3068  cun 3945  cin 3946  wss 3947  {csn 4627  cop 4633  dom cdm 5675  cres 5677  Rel wrel 5680   Fn wfn 6537  cfv 6542   predc-bnj14 33997   FrSe w-bnj15 34001   trClc-bnj18 34003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-bnj17 33996  df-bnj14 33998  df-bnj13 34000  df-bnj15 34002  df-bnj18 34004  df-bnj19 34006
This theorem is referenced by:  bnj1489  34365
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