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Theorem fnejoin2 34697
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 4874 . . . . . . . . 9 (𝑋𝑉 {𝑋} = 𝑋)
21eqcomd 2742 . . . . . . . 8 (𝑋𝑉𝑋 = {𝑋})
32adantr 481 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → 𝑋 = {𝑋})
4 iftrue 4480 . . . . . . . . 9 (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = {𝑋})
54unieqd 4867 . . . . . . . 8 (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = {𝑋})
65eqeq2d 2747 . . . . . . 7 (𝑆 = ∅ → (𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆) ↔ 𝑋 = {𝑋}))
73, 6syl5ibrcom 246 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑆 = ∅ → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
8 n0 4294 . . . . . . 7 (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥𝑆)
9 unieq 4864 . . . . . . . . . . . . 13 (𝑦 = 𝑥 𝑦 = 𝑥)
109eqeq2d 2747 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑋 = 𝑦𝑋 = 𝑥))
1110rspccva 3569 . . . . . . . . . . 11 ((∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = 𝑥)
12113adant1 1129 . . . . . . . . . 10 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = 𝑥)
13 fnejoin1 34696 . . . . . . . . . . 11 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆))
14 eqid 2736 . . . . . . . . . . . 12 𝑥 = 𝑥
15 eqid 2736 . . . . . . . . . . . 12 if(𝑆 = ∅, {𝑋}, 𝑆) = if(𝑆 = ∅, {𝑋}, 𝑆)
1614, 15fnebas 34672 . . . . . . . . . . 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 2776 . . . . . . . . 9 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
19183expia 1120 . . . . . . . 8 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑥𝑆𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
2019exlimdv 1935 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (∃𝑥 𝑥𝑆𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
218, 20biimtrid 241 . . . . . 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 2736 . . . . . 6 𝑇 = 𝑇
2415, 23fnebas 34672 . . . . 5 (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇)
2522, 24sylan9eq 2796 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇) → 𝑋 = 𝑇)
2625ex 413 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑋 = 𝑇))
27 fnetr 34679 . . . . . . 7 ((𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆) ∧ if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇) → 𝑥Fne𝑇)
2827ex 413 . . . . . 6 (𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
2913, 28syl 17 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
30293expa 1117 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ 𝑥𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
3130ralrimdva 3147 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 → ∀𝑥𝑆 𝑥Fne𝑇))
3226, 31jcad 513 . 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 768 . . . . 5 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 = 𝑇)
3533, 34eqtr3d 2778 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇)
36 sseq1 3957 . . . . 5 ({𝑋} = if(𝑆 = ∅, {𝑋}, 𝑆) → ({𝑋} ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
37 sseq1 3957 . . . . 5 ( 𝑆 = if(𝑆 = ∅, {𝑋}, 𝑆) → ( 𝑆 ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
38 elex 3459 . . . . . . . . . . . 12 (𝑋𝑉𝑋 ∈ V)
3938ad2antrr 723 . . . . . . . . . . 11 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 ∈ V)
4034, 39eqeltrrd 2838 . . . . . . . . . 10 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
41 uniexb 7677 . . . . . . . . . 10 (𝑇 ∈ V ↔ 𝑇 ∈ V)
4240, 41sylibr 233 . . . . . . . . 9 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
43 ssid 3954 . . . . . . . . 9 𝑇𝑇
44 eltg3i 22218 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑇𝑇) → 𝑇 ∈ (topGen‘𝑇))
4542, 43, 44sylancl 586 . . . . . . . 8 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ (topGen‘𝑇))
4634, 45eqeltrd 2837 . . . . . . 7 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 ∈ (topGen‘𝑇))
4746snssd 4757 . . . . . 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 775 . . . . . . 7 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥𝑆 𝑥Fne𝑇)
50 fnetg 34673 . . . . . . . 8 (𝑥Fne𝑇𝑥 ⊆ (topGen‘𝑇))
5150ralimi 3082 . . . . . . 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 4888 . . . . . 6 ( 𝑆 ⊆ (topGen‘𝑇) ↔ ∀𝑥𝑆 𝑥 ⊆ (topGen‘𝑇))
5452, 53sylibr 233 . . . . 5 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → 𝑆 ⊆ (topGen‘𝑇))
5536, 37, 48, 54ifbothda 4512 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇))
5615, 23isfne4 34668 . . . 4 (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 ↔ ( if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇 ∧ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
5735, 55, 56sylanbrc 583 . . 3 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇)
5857ex 413 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → ((𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇) → if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇))
5932, 58impbid 211 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 205  wa 396  w3a 1086   = wceq 1540  wex 1780  wcel 2105  wne 2940  wral 3061  Vcvv 3441  wss 3898  c0 4270  ifcif 4474  {csn 4574   cuni 4853   class class class wbr 5093  cfv 6480  topGenctg 17246  Fnecfne 34664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6432  df-fun 6482  df-fv 6488  df-topgen 17252  df-fne 34665
This theorem is referenced by: (None)
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