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Theorem fnejoin2 32041
Description: Join of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
fnejoin2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 ↔ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)))
Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝑉   𝑥,𝑋,𝑦   𝑥,𝑇
Allowed substitution hints:   𝑇(𝑦)   𝑉(𝑦)

Proof of Theorem fnejoin2
StepHypRef Expression
1 unisng 4423 . . . . . . . . 9 (𝑋𝑉 {𝑋} = 𝑋)
21eqcomd 2627 . . . . . . . 8 (𝑋𝑉𝑋 = {𝑋})
32adantr 481 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → 𝑋 = {𝑋})
4 iftrue 4069 . . . . . . . . 9 (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = {𝑋})
54unieqd 4417 . . . . . . . 8 (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = {𝑋})
65eqeq2d 2631 . . . . . . 7 (𝑆 = ∅ → (𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆) ↔ 𝑋 = {𝑋}))
73, 6syl5ibrcom 237 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑆 = ∅ → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
8 n0 3912 . . . . . . 7 (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥𝑆)
9 unieq 4415 . . . . . . . . . . . . 13 (𝑦 = 𝑥 𝑦 = 𝑥)
109eqeq2d 2631 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑋 = 𝑦𝑋 = 𝑥))
1110rspccva 3297 . . . . . . . . . . 11 ((∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = 𝑥)
12113adant1 1077 . . . . . . . . . 10 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = 𝑥)
13 fnejoin1 32040 . . . . . . . . . . 11 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆))
14 eqid 2621 . . . . . . . . . . . 12 𝑥 = 𝑥
15 eqid 2621 . . . . . . . . . . . 12 if(𝑆 = ∅, {𝑋}, 𝑆) = if(𝑆 = ∅, {𝑋}, 𝑆)
1614, 15fnebas 32016 . . . . . . . . . . 11 (𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆) → 𝑥 = if(𝑆 = ∅, {𝑋}, 𝑆))
1713, 16syl 17 . . . . . . . . . 10 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑥 = if(𝑆 = ∅, {𝑋}, 𝑆))
1812, 17eqtrd 2655 . . . . . . . . 9 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
19183expia 1264 . . . . . . . 8 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑥𝑆𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
2019exlimdv 1858 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (∃𝑥 𝑥𝑆𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
218, 20syl5bi 232 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑆 ≠ ∅ → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
227, 21pm2.61dne 2876 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
23 eqid 2621 . . . . . 6 𝑇 = 𝑇
2415, 23fnebas 32016 . . . . 5 (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇)
2522, 24sylan9eq 2675 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇) → 𝑋 = 𝑇)
2625ex 450 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑋 = 𝑇))
27 fnetr 32023 . . . . . . 7 ((𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆) ∧ if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇) → 𝑥Fne𝑇)
2827ex 450 . . . . . 6 (𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
2913, 28syl 17 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
30293expa 1262 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ 𝑥𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
3130ralrimdva 2964 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 → ∀𝑥𝑆 𝑥Fne𝑇))
3226, 31jcad 555 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 → (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)))
3322adantr 481 . . . . 5 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
34 simprl 793 . . . . 5 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 = 𝑇)
3533, 34eqtr3d 2657 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇)
36 sseq1 3610 . . . . 5 ({𝑋} = if(𝑆 = ∅, {𝑋}, 𝑆) → ({𝑋} ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
37 sseq1 3610 . . . . 5 ( 𝑆 = if(𝑆 = ∅, {𝑋}, 𝑆) → ( 𝑆 ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
38 elex 3201 . . . . . . . . . . . 12 (𝑋𝑉𝑋 ∈ V)
3938ad2antrr 761 . . . . . . . . . . 11 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 ∈ V)
4034, 39eqeltrrd 2699 . . . . . . . . . 10 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
41 uniexb 6928 . . . . . . . . . 10 (𝑇 ∈ V ↔ 𝑇 ∈ V)
4240, 41sylibr 224 . . . . . . . . 9 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
43 ssid 3608 . . . . . . . . 9 𝑇𝑇
44 eltg3i 20689 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑇𝑇) → 𝑇 ∈ (topGen‘𝑇))
4542, 43, 44sylancl 693 . . . . . . . 8 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ (topGen‘𝑇))
4634, 45eqeltrd 2698 . . . . . . 7 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 ∈ (topGen‘𝑇))
4746snssd 4314 . . . . . 6 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → {𝑋} ⊆ (topGen‘𝑇))
4847adantr 481 . . . . 5 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ 𝑆 = ∅) → {𝑋} ⊆ (topGen‘𝑇))
49 simplrr 800 . . . . . . 7 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥𝑆 𝑥Fne𝑇)
50 fnetg 32017 . . . . . . . 8 (𝑥Fne𝑇𝑥 ⊆ (topGen‘𝑇))
5150ralimi 2947 . . . . . . 7 (∀𝑥𝑆 𝑥Fne𝑇 → ∀𝑥𝑆 𝑥 ⊆ (topGen‘𝑇))
5249, 51syl 17 . . . . . 6 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥𝑆 𝑥 ⊆ (topGen‘𝑇))
53 unissb 4440 . . . . . 6 ( 𝑆 ⊆ (topGen‘𝑇) ↔ ∀𝑥𝑆 𝑥 ⊆ (topGen‘𝑇))
5452, 53sylibr 224 . . . . 5 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → 𝑆 ⊆ (topGen‘𝑇))
5536, 37, 48, 54ifbothda 4100 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇))
5615, 23isfne4 32012 . . . 4 (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 ↔ ( if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇 ∧ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
5735, 55, 56sylanbrc 697 . . 3 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇)
5857ex 450 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → ((𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇) → if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇))
5932, 58impbid 202 1 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 ↔ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  Vcvv 3189  wss 3559  c0 3896  ifcif 4063  {csn 4153   cuni 4407   class class class wbr 4618  cfv 5852  topGenctg 16030  Fnecfne 32008
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-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  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-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-topgen 16036  df-fne 32009
This theorem is referenced by: (None)
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