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Theorem fnejoin2 32694
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 4590 . . . . . . . . 9 (𝑋𝑉 {𝑋} = 𝑋)
21eqcomd 2777 . . . . . . . 8 (𝑋𝑉𝑋 = {𝑋})
32adantr 466 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → 𝑋 = {𝑋})
4 iftrue 4231 . . . . . . . . 9 (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = {𝑋})
54unieqd 4584 . . . . . . . 8 (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = {𝑋})
65eqeq2d 2781 . . . . . . 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 4078 . . . . . . 7 (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥𝑆)
9 unieq 4582 . . . . . . . . . . . . 13 (𝑦 = 𝑥 𝑦 = 𝑥)
109eqeq2d 2781 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑋 = 𝑦𝑋 = 𝑥))
1110rspccva 3459 . . . . . . . . . . 11 ((∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = 𝑥)
12113adant1 1124 . . . . . . . . . 10 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = 𝑥)
13 fnejoin1 32693 . . . . . . . . . . 11 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆))
14 eqid 2771 . . . . . . . . . . . 12 𝑥 = 𝑥
15 eqid 2771 . . . . . . . . . . . 12 if(𝑆 = ∅, {𝑋}, 𝑆) = if(𝑆 = ∅, {𝑋}, 𝑆)
1614, 15fnebas 32669 . . . . . . . . . . 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 2805 . . . . . . . . 9 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
19183expia 1114 . . . . . . . 8 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑥𝑆𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆)))
2019exlimdv 2013 . . . . . . 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 3029 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
23 eqid 2771 . . . . . 6 𝑇 = 𝑇
2415, 23fnebas 32669 . . . . 5 (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇)
2522, 24sylan9eq 2825 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇) → 𝑋 = 𝑇)
2625ex 397 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑋 = 𝑇))
27 fnetr 32676 . . . . . . 7 ((𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆) ∧ if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇) → 𝑥Fne𝑇)
2827ex 397 . . . . . 6 (𝑥Fneif(𝑆 = ∅, {𝑋}, 𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
2913, 28syl 17 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝑥𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
30293expa 1111 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ 𝑥𝑆) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇𝑥Fne𝑇))
3130ralrimdva 3118 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 → ∀𝑥𝑆 𝑥Fne𝑇))
3226, 31jcad 502 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 → (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)))
3322adantr 466 . . . . 5 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 = if(𝑆 = ∅, {𝑋}, 𝑆))
34 simprl 754 . . . . 5 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 = 𝑇)
3533, 34eqtr3d 2807 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇)
36 sseq1 3775 . . . . 5 ({𝑋} = if(𝑆 = ∅, {𝑋}, 𝑆) → ({𝑋} ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
37 sseq1 3775 . . . . 5 ( 𝑆 = if(𝑆 = ∅, {𝑋}, 𝑆) → ( 𝑆 ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
38 elex 3364 . . . . . . . . . . . 12 (𝑋𝑉𝑋 ∈ V)
3938ad2antrr 705 . . . . . . . . . . 11 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 ∈ V)
4034, 39eqeltrrd 2851 . . . . . . . . . 10 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
41 uniexb 7118 . . . . . . . . . 10 (𝑇 ∈ V ↔ 𝑇 ∈ V)
4240, 41sylibr 224 . . . . . . . . 9 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
43 ssid 3773 . . . . . . . . 9 𝑇𝑇
44 eltg3i 20979 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑇𝑇) → 𝑇 ∈ (topGen‘𝑇))
4542, 43, 44sylancl 574 . . . . . . . 8 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑇 ∈ (topGen‘𝑇))
4634, 45eqeltrd 2850 . . . . . . 7 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → 𝑋 ∈ (topGen‘𝑇))
4746snssd 4475 . . . . . 6 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → {𝑋} ⊆ (topGen‘𝑇))
4847adantr 466 . . . . 5 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ 𝑆 = ∅) → {𝑋} ⊆ (topGen‘𝑇))
49 simplrr 763 . . . . . . 7 ((((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥𝑆 𝑥Fne𝑇)
50 fnetg 32670 . . . . . . . 8 (𝑥Fne𝑇𝑥 ⊆ (topGen‘𝑇))
5150ralimi 3101 . . . . . . 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 4605 . . . . . 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 4262 . . . 4 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇))
5615, 23isfne4 32665 . . . 4 (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 ↔ ( if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑇 ∧ if(𝑆 = ∅, {𝑋}, 𝑆) ⊆ (topGen‘𝑇)))
5735, 55, 56sylanbrc 572 . . 3 (((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) ∧ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇)
5857ex 397 . 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 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  Vcvv 3351  wss 3723  c0 4063  ifcif 4225  {csn 4316   cuni 4574   class class class wbr 4786  cfv 6029  topGenctg 16299  Fnecfne 32661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5992  df-fun 6031  df-fv 6037  df-topgen 16305  df-fne 32662
This theorem is referenced by: (None)
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