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Theorem fnejoin2 34295
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 4840 . . . . . . . . 9 (𝑋𝑉 {𝑋} = 𝑋)
21eqcomd 2743 . . . . . . . 8 (𝑋𝑉𝑋 = {𝑋})
32adantr 484 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → 𝑋 = {𝑋})
4 iftrue 4445 . . . . . . . . 9 (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = {𝑋})
54unieqd 4833 . . . . . . . 8 (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = {𝑋})
65eqeq2d 2748 . . . . . . 7 (𝑆 = ∅ → (𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆) ↔ 𝑋 = {𝑋}))
73, 6syl5ibrcom 250 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑆 = ∅ → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
8 n0 4261 . . . . . . 7 (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥𝑆)
9 unieq 4830 . . . . . . . . . . . . 13 (𝑦 = 𝑥 𝑦 = 𝑥)
109eqeq2d 2748 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑋 = 𝑦𝑋 = 𝑥))
1110rspccva 3536 . . . . . . . . . . 11 ((∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = 𝑥)
12113adant1 1132 . . . . . . . . . 10 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = 𝑥)
13 fnejoin1 34294 . . . . . . . . . . 11 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆))
14 eqid 2737 . . . . . . . . . . . 12 𝑥 = 𝑥
15 eqid 2737 . . . . . . . . . . . 12 if(𝑆 = ∅, {𝑋}, 𝑆) = if(𝑆 = ∅, {𝑋}, 𝑆)
1614, 15fnebas 34270 . . . . . . . . . . 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 2777 . . . . . . . . 9 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
19183expia 1123 . . . . . . . 8 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑥𝑆𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
2019exlimdv 1941 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (∃𝑥 𝑥𝑆𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
218, 20syl5bi 245 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑆 ≠ ∅ → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
227, 21pm2.61dne 3028 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
23 eqid 2737 . . . . . 6 𝑇 = 𝑇
2415, 23fnebas 34270 . . . . 5 (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇)
2522, 24sylan9eq 2798 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇) → 𝑋 = 𝑇)
2625ex 416 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑋 = 𝑇))
27 fnetr 34277 . . . . . . 7 ((𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆) ∧ if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇) → 𝑥Fne𝑇)
2827ex 416 . . . . . 6 (𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
2913, 28syl 17 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
30293expa 1120 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ 𝑥𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
3130ralrimdva 3110 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 → ∀𝑥𝑆 𝑥Fne𝑇))
3226, 31jcad 516 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 → (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)))
3322adantr 484 . . . . 5 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
34 simprl 771 . . . . 5 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 = 𝑇)
3533, 34eqtr3d 2779 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇)
36 sseq1 3926 . . . . 5 ({𝑋} = if(𝑆 = ∅, {𝑋}, 𝑆) → ({𝑋} ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
37 sseq1 3926 . . . . 5 ( 𝑆 = if(𝑆 = ∅, {𝑋}, 𝑆) → ( 𝑆 ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
38 elex 3426 . . . . . . . . . . . 12 (𝑋𝑉𝑋 ∈ V)
3938ad2antrr 726 . . . . . . . . . . 11 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 ∈ V)
4034, 39eqeltrrd 2839 . . . . . . . . . 10 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
41 uniexb 7549 . . . . . . . . . 10 (𝑇 ∈ V ↔ 𝑇 ∈ V)
4240, 41sylibr 237 . . . . . . . . 9 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
43 ssid 3923 . . . . . . . . 9 𝑇𝑇
44 eltg3i 21858 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑇𝑇) → 𝑇 ∈ (topGen‘𝑇))
4542, 43, 44sylancl 589 . . . . . . . 8 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ (topGen‘𝑇))
4634, 45eqeltrd 2838 . . . . . . 7 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 ∈ (topGen‘𝑇))
4746snssd 4722 . . . . . 6 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → {𝑋} ⊆ (topGen‘𝑇))
4847adantr 484 . . . . 5 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ 𝑆 = ∅) → {𝑋} ⊆ (topGen‘𝑇))
49 simplrr 778 . . . . . . 7 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥𝑆 𝑥Fne𝑇)
50 fnetg 34271 . . . . . . . 8 (𝑥Fne𝑇𝑥 ⊆ (topGen‘𝑇))
5150ralimi 3083 . . . . . . 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 4853 . . . . . 6 ( 𝑆 ⊆ (topGen‘𝑇) ↔ ∀𝑥𝑆 𝑥 ⊆ (topGen‘𝑇))
5452, 53sylibr 237 . . . . 5 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → 𝑆 ⊆ (topGen‘𝑇))
5536, 37, 48, 54ifbothda 4477 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇))
5615, 23isfne4 34266 . . . 4 (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 ↔ ( if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇 ∧ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
5735, 55, 56sylanbrc 586 . . 3 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇)
5857ex 416 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → ((𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇) → if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇))
5932, 58impbid 215 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 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  wne 2940  wral 3061  Vcvv 3408  wss 3866  c0 4237  ifcif 4439  {csn 4541   cuni 4819   class class class wbr 5053  cfv 6380  topGenctg 16942  Fnecfne 34262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-topgen 16948  df-fne 34263
This theorem is referenced by: (None)
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