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Theorem qtopval2 23421
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 1135 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽𝑉)
2 fof 6805 . . . . 5 (𝐹:𝑍onto𝑌𝐹:𝑍𝑌)
323ad2ant2 1133 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹:𝑍𝑌)
4 qtopval.1 . . . . . 6 𝑋 = 𝐽
5 uniexg 7734 . . . . . . 7 (𝐽𝑉 𝐽 ∈ V)
653ad2ant1 1132 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽 ∈ V)
74, 6eqeltrid 2836 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑋 ∈ V)
8 simp3 1137 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍𝑋)
97, 8ssexd 5324 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍 ∈ V)
103, 9fexd 7231 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹 ∈ V)
114qtopval 23420 . . 3 ((𝐽𝑉𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
121, 10, 11syl2anc 583 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
13 imassrn 6070 . . . . . 6 (𝐹𝑋) ⊆ ran 𝐹
14 forn 6808 . . . . . . 7 (𝐹:𝑍onto𝑌 → ran 𝐹 = 𝑌)
15143ad2ant2 1133 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ran 𝐹 = 𝑌)
1613, 15sseqtrid 4034 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) ⊆ 𝑌)
17 foima 6810 . . . . . . 7 (𝐹:𝑍onto𝑌 → (𝐹𝑍) = 𝑌)
18173ad2ant2 1133 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) = 𝑌)
19 imass2 6101 . . . . . . 7 (𝑍𝑋 → (𝐹𝑍) ⊆ (𝐹𝑋))
208, 19syl 17 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) ⊆ (𝐹𝑋))
2118, 20eqsstrrd 4021 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑌 ⊆ (𝐹𝑋))
2216, 21eqssd 3999 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) = 𝑌)
2322pweqd 4619 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝒫 (𝐹𝑋) = 𝒫 𝑌)
24 cnvimass 6080 . . . . . . 7 (𝐹𝑠) ⊆ dom 𝐹
2524, 3fssdm 6737 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑍)
2625, 8sstrd 3992 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑋)
27 df-ss 3965 . . . . 5 ((𝐹𝑠) ⊆ 𝑋 ↔ ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
2826, 27sylib 217 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
2928eleq1d 2817 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (((𝐹𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (𝐹𝑠) ∈ 𝐽))
3023, 29rabeqbidv 3448 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
3112, 30eqtrd 2771 1 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1540  wcel 2105  {crab 3431  Vcvv 3473  cin 3947  wss 3948  𝒫 cpw 4602   cuni 4908  ccnv 5675  ran crn 5677  cima 5679  wf 6539  ontowfo 6541  (class class class)co 7412   qTop cqtop 17454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-qtop 17458
This theorem is referenced by:  elqtop  23422
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