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Theorem fnejoin1 31997
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 4438 . . . . . 6 (𝐴𝑆𝐴 𝑆)
213ad2ant3 1082 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 𝑆)
32unissd 4433 . . . 4 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 𝑆)
4 eqimss2 3642 . . . . . . . . . 10 (𝑋 = 𝑦 𝑦𝑋)
5 sspwuni 4582 . . . . . . . . . 10 (𝑦 ⊆ 𝒫 𝑋 𝑦𝑋)
64, 5sylibr 224 . . . . . . . . 9 (𝑋 = 𝑦𝑦 ⊆ 𝒫 𝑋)
76ralimi 2952 . . . . . . . 8 (∀𝑦𝑆 𝑋 = 𝑦 → ∀𝑦𝑆 𝑦 ⊆ 𝒫 𝑋)
873ad2ant2 1081 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → ∀𝑦𝑆 𝑦 ⊆ 𝒫 𝑋)
9 unissb 4440 . . . . . . 7 ( 𝑆 ⊆ 𝒫 𝑋 ↔ ∀𝑦𝑆 𝑦 ⊆ 𝒫 𝑋)
108, 9sylibr 224 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ⊆ 𝒫 𝑋)
11 sspwuni 4582 . . . . . 6 ( 𝑆 ⊆ 𝒫 𝑋 𝑆𝑋)
1210, 11sylib 208 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆𝑋)
13 unieq 4415 . . . . . . . 8 (𝑦 = 𝐴 𝑦 = 𝐴)
1413eqeq2d 2636 . . . . . . 7 (𝑦 = 𝐴 → (𝑋 = 𝑦𝑋 = 𝐴))
1514rspccva 3299 . . . . . 6 ((∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑋 = 𝐴)
16153adant1 1077 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑋 = 𝐴)
1712, 16sseqtrd 3625 . . . 4 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 𝐴)
183, 17eqssd 3605 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 = 𝑆)
19 pwexg 4815 . . . . . . 7 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
20193ad2ant1 1080 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝒫 𝑋 ∈ V)
2120, 10ssexd 4770 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ∈ V)
22 bastg 20676 . . . . 5 ( 𝑆 ∈ V → 𝑆 ⊆ (topGen‘ 𝑆))
2321, 22syl 17 . . . 4 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ⊆ (topGen‘ 𝑆))
242, 23sstrd 3598 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 ⊆ (topGen‘ 𝑆))
25 eqid 2626 . . . 4 𝐴 = 𝐴
26 eqid 2626 . . . 4 𝑆 = 𝑆
2725, 26isfne4 31969 . . 3 (𝐴Fne 𝑆 ↔ ( 𝐴 = 𝑆𝐴 ⊆ (topGen‘ 𝑆)))
2818, 24, 27sylanbrc 697 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴Fne 𝑆)
29 ne0i 3902 . . . 4 (𝐴𝑆𝑆 ≠ ∅)
30293ad2ant3 1082 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ≠ ∅)
31 ifnefalse 4075 . . 3 (𝑆 ≠ ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑆)
3230, 31syl 17 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑆)
3328, 32breqtrrd 4646 1 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  Vcvv 3191  wss 3560  c0 3896  ifcif 4063  𝒫 cpw 4135  {csn 4153   cuni 4407   class class class wbr 4618  cfv 5850  topGenctg 16014  Fnecfne 31965
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  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 5813  df-fun 5852  df-fv 5858  df-topgen 16020  df-fne 31966
This theorem is referenced by:  fnejoin2  31998
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