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Theorem fnejoin1 32488
Description: Join of equivalence classes under the fineness relation-part one. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
fnejoin1 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, 𝑆))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑆   𝑦,𝑋
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem fnejoin1
StepHypRef Expression
1 elssuni 4499 . . . . . 6 (𝐴𝑆𝐴 𝑆)
213ad2ant3 1104 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 𝑆)
32unissd 4494 . . . 4 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 𝑆)
4 eqimss2 3691 . . . . . . . . . 10 (𝑋 = 𝑦 𝑦𝑋)
5 sspwuni 4643 . . . . . . . . . 10 (𝑦 ⊆ 𝒫 𝑋 𝑦𝑋)
64, 5sylibr 224 . . . . . . . . 9 (𝑋 = 𝑦𝑦 ⊆ 𝒫 𝑋)
76ralimi 2981 . . . . . . . 8 (∀𝑦𝑆 𝑋 = 𝑦 → ∀𝑦𝑆 𝑦 ⊆ 𝒫 𝑋)
873ad2ant2 1103 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → ∀𝑦𝑆 𝑦 ⊆ 𝒫 𝑋)
9 unissb 4501 . . . . . . 7 ( 𝑆 ⊆ 𝒫 𝑋 ↔ ∀𝑦𝑆 𝑦 ⊆ 𝒫 𝑋)
108, 9sylibr 224 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ⊆ 𝒫 𝑋)
11 sspwuni 4643 . . . . . 6 ( 𝑆 ⊆ 𝒫 𝑋 𝑆𝑋)
1210, 11sylib 208 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆𝑋)
13 unieq 4476 . . . . . . . 8 (𝑦 = 𝐴 𝑦 = 𝐴)
1413eqeq2d 2661 . . . . . . 7 (𝑦 = 𝐴 → (𝑋 = 𝑦𝑋 = 𝐴))
1514rspccva 3339 . . . . . 6 ((∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑋 = 𝐴)
16153adant1 1099 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑋 = 𝐴)
1712, 16sseqtrd 3674 . . . 4 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 𝐴)
183, 17eqssd 3653 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 = 𝑆)
19 pwexg 4880 . . . . . . 7 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
20193ad2ant1 1102 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝒫 𝑋 ∈ V)
2120, 10ssexd 4838 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ∈ V)
22 bastg 20818 . . . . 5 ( 𝑆 ∈ V → 𝑆 ⊆ (topGen‘ 𝑆))
2321, 22syl 17 . . . 4 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ⊆ (topGen‘ 𝑆))
242, 23sstrd 3646 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 ⊆ (topGen‘ 𝑆))
25 eqid 2651 . . . 4 𝐴 = 𝐴
26 eqid 2651 . . . 4 𝑆 = 𝑆
2725, 26isfne4 32460 . . 3 (𝐴Fne 𝑆 ↔ ( 𝐴 = 𝑆𝐴 ⊆ (topGen‘ 𝑆)))
2818, 24, 27sylanbrc 699 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴Fne 𝑆)
29 ne0i 3954 . . . 4 (𝐴𝑆𝑆 ≠ ∅)
30293ad2ant3 1104 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ≠ ∅)
31 ifnefalse 4131 . . 3 (𝑆 ≠ ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑆)
3230, 31syl 17 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑆)
3328, 32breqtrrd 4713 1 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210   cuni 4468   class class class wbr 4685  cfv 5926  topGenctg 16145  Fnecfne 32456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151  df-fne 32457
This theorem is referenced by:  fnejoin2  32489
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