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Theorem bnj1321 33639
Description: Technical lemma for bnj60 33674. 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 485 . 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 498 . . . . . . . 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 2909 . . . . . . . . . . . . 13 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
86, 7nfcxfr 2905 . . . . . . . . . . . 12 𝑓𝐶
98nfcri 2894 . . . . . . . . . . 11 𝑓 𝑔𝐶
10 nfv 1917 . . . . . . . . . . 11 𝑓dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10nfan 1902 . . . . . . . . . 10 𝑓(𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12 eleq1w 2820 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
13 dmeq 5859 . . . . . . . . . . . . 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 631 . . . . . . . . . . 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 2308 . . . . . . . . 9 ([𝑔 / 𝑓]𝜏 ↔ (𝑔𝐶 ∧ dom 𝑔 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
1817simplbi 498 . . . . . . . 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 33638 . . . . . . 7 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
242, 5, 19, 23syl3anc 1371 . . . . . 6 ((𝑅 FrSe 𝐴𝜏 ∧ [𝑔 / 𝑓]𝜏) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
253simprbi 497 . . . . . . . . . 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 497 . . . . . . . . . 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 33434 . . . . . . . . 9 (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑓)
3130reseq2d 5937 . . . . . . . 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 5732 . . . . . . . . 9 (𝑧 = 𝑓 → (Rel 𝑧 ↔ Rel 𝑓))
3420, 21, 6bnj66 33472 . . . . . . . . 9 (𝑧𝐶 → Rel 𝑧)
3533, 34vtoclga 3534 . . . . . . . 8 (𝑓𝐶 → Rel 𝑓)
36 resdm 5982 . . . . . . . 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 231 . . . . . . . . 9 (dom 𝑓 = dom 𝑔 → (dom 𝑓 ∩ dom 𝑔) = dom 𝑔)
4140reseq2d 5937 . . . . . . . 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 33472 . . . . . . . 8 (𝑔𝐶 → Rel 𝑔)
44 resdm 5982 . . . . . . . 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 1931 . . 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 1917 . . 3 𝑔𝜏
5251eu2 2609 . 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 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wex 1781  [wsb 2067  wcel 2106  ∃!weu 2566  {cab 2713  wral 3064  wrex 3073  cun 3908  cin 3909  wss 3910  {csn 4586  cop 4592  dom cdm 5633  cres 5635  Rel wrel 5638   Fn wfn 6491  cfv 6496   predc-bnj14 33300   FrSe w-bnj15 33304   trClc-bnj18 33306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-bnj17 33299  df-bnj14 33301  df-bnj13 33303  df-bnj15 33305  df-bnj18 33307  df-bnj19 33309
This theorem is referenced by:  bnj1489  33668
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