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Theorem fnejoin1 33830
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 4833 . . . . . 6 (𝐴𝑆𝐴 𝑆)
213ad2ant3 1132 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 𝑆)
32unissd 4813 . . . 4 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 𝑆)
4 eqimss2 3975 . . . . . . . . . 10 (𝑋 = 𝑦 𝑦𝑋)
5 sspwuni 4988 . . . . . . . . . 10 (𝑦 ⊆ 𝒫 𝑋 𝑦𝑋)
64, 5sylibr 237 . . . . . . . . 9 (𝑋 = 𝑦𝑦 ⊆ 𝒫 𝑋)
76ralimi 3131 . . . . . . . 8 (∀𝑦𝑆 𝑋 = 𝑦 → ∀𝑦𝑆 𝑦 ⊆ 𝒫 𝑋)
873ad2ant2 1131 . . . . . . 7 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → ∀𝑦𝑆 𝑦 ⊆ 𝒫 𝑋)
9 unissb 4835 . . . . . . 7 ( 𝑆 ⊆ 𝒫 𝑋 ↔ ∀𝑦𝑆 𝑦 ⊆ 𝒫 𝑋)
108, 9sylibr 237 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ⊆ 𝒫 𝑋)
11 sspwuni 4988 . . . . . 6 ( 𝑆 ⊆ 𝒫 𝑋 𝑆𝑋)
1210, 11sylib 221 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆𝑋)
13 unieq 4814 . . . . . . . 8 (𝑦 = 𝐴 𝑦 = 𝐴)
1413eqeq2d 2812 . . . . . . 7 (𝑦 = 𝐴 → (𝑋 = 𝑦𝑋 = 𝐴))
1514rspccva 3573 . . . . . 6 ((∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑋 = 𝐴)
16153adant1 1127 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑋 = 𝐴)
1712, 16sseqtrd 3958 . . . 4 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 𝐴)
183, 17eqssd 3935 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 = 𝑆)
19 pwexg 5247 . . . . . . 7 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
20193ad2ant1 1130 . . . . . 6 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝒫 𝑋 ∈ V)
2120, 10ssexd 5195 . . . . 5 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ∈ V)
22 bastg 21575 . . . . 5 ( 𝑆 ∈ V → 𝑆 ⊆ (topGen‘ 𝑆))
2321, 22syl 17 . . . 4 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ⊆ (topGen‘ 𝑆))
242, 23sstrd 3928 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴 ⊆ (topGen‘ 𝑆))
25 eqid 2801 . . . 4 𝐴 = 𝐴
26 eqid 2801 . . . 4 𝑆 = 𝑆
2725, 26isfne4 33802 . . 3 (𝐴Fne 𝑆 ↔ ( 𝐴 = 𝑆𝐴 ⊆ (topGen‘ 𝑆)))
2818, 24, 27sylanbrc 586 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴Fne 𝑆)
29 ne0i 4253 . . . 4 (𝐴𝑆𝑆 ≠ ∅)
30293ad2ant3 1132 . . 3 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝑆 ≠ ∅)
31 ifnefalse 4440 . . 3 (𝑆 ≠ ∅ → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑆)
3230, 31syl 17 . 2 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → if(𝑆 = ∅, {𝑋}, 𝑆) = 𝑆)
3328, 32breqtrrd 5061 1 ((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  Vcvv 3444  wss 3884  c0 4246  ifcif 4428  𝒫 cpw 4500  {csn 4528   cuni 4803   class class class wbr 5033  cfv 6328  topGenctg 16707  Fnecfne 33798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-topgen 16713  df-fne 33799
This theorem is referenced by:  fnejoin2  33831
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