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Theorem qtopval2 23704
Description: Value of the quotient topology function when 𝐹 is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
qtopval.1 𝑋 = 𝐽
Assertion
Ref Expression
qtopval2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
Distinct variable groups:   𝐹,𝑠   𝐽,𝑠   𝑉,𝑠   𝑌,𝑠   𝑍,𝑠   𝑋,𝑠

Proof of Theorem qtopval2
StepHypRef Expression
1 simp1 1137 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽𝑉)
2 fof 6820 . . . . 5 (𝐹:𝑍onto𝑌𝐹:𝑍𝑌)
323ad2ant2 1135 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹:𝑍𝑌)
4 qtopval.1 . . . . . 6 𝑋 = 𝐽
5 uniexg 7760 . . . . . . 7 (𝐽𝑉 𝐽 ∈ V)
653ad2ant1 1134 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽 ∈ V)
74, 6eqeltrid 2845 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑋 ∈ V)
8 simp3 1139 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍𝑋)
97, 8ssexd 5324 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍 ∈ V)
103, 9fexd 7247 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹 ∈ V)
114qtopval 23703 . . 3 ((𝐽𝑉𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
121, 10, 11syl2anc 584 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
13 imassrn 6089 . . . . . 6 (𝐹𝑋) ⊆ ran 𝐹
14 forn 6823 . . . . . . 7 (𝐹:𝑍onto𝑌 → ran 𝐹 = 𝑌)
15143ad2ant2 1135 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ran 𝐹 = 𝑌)
1613, 15sseqtrid 4026 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) ⊆ 𝑌)
17 foima 6825 . . . . . . 7 (𝐹:𝑍onto𝑌 → (𝐹𝑍) = 𝑌)
18173ad2ant2 1135 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) = 𝑌)
19 imass2 6120 . . . . . . 7 (𝑍𝑋 → (𝐹𝑍) ⊆ (𝐹𝑋))
208, 19syl 17 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) ⊆ (𝐹𝑋))
2118, 20eqsstrrd 4019 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑌 ⊆ (𝐹𝑋))
2216, 21eqssd 4001 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) = 𝑌)
2322pweqd 4617 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝒫 (𝐹𝑋) = 𝒫 𝑌)
24 cnvimass 6100 . . . . . . 7 (𝐹𝑠) ⊆ dom 𝐹
2524, 3fssdm 6755 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑍)
2625, 8sstrd 3994 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑋)
27 dfss2 3969 . . . . 5 ((𝐹𝑠) ⊆ 𝑋 ↔ ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
2826, 27sylib 218 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
2928eleq1d 2826 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (((𝐹𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (𝐹𝑠) ∈ 𝐽))
3023, 29rabeqbidv 3455 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
3112, 30eqtrd 2777 1 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907  ccnv 5684  ran crn 5686  cima 5688  wf 6557  ontowfo 6559  (class class class)co 7431   qTop cqtop 17548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17552
This theorem is referenced by:  elqtop  23705
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