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Theorem qtopval2 21409
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 1059 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽𝑉)
2 fof 6072 . . . . 5 (𝐹:𝑍onto𝑌𝐹:𝑍𝑌)
323ad2ant2 1081 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹:𝑍𝑌)
4 qtopval.1 . . . . . 6 𝑋 = 𝐽
5 uniexg 6908 . . . . . . 7 (𝐽𝑉 𝐽 ∈ V)
653ad2ant1 1080 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽 ∈ V)
74, 6syl5eqel 2702 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑋 ∈ V)
8 simp3 1061 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍𝑋)
97, 8ssexd 4765 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍 ∈ V)
10 fex 6444 . . . 4 ((𝐹:𝑍𝑌𝑍 ∈ V) → 𝐹 ∈ V)
113, 9, 10syl2anc 692 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹 ∈ V)
124qtopval 21408 . . 3 ((𝐽𝑉𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
131, 11, 12syl2anc 692 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
14 imassrn 5436 . . . . . 6 (𝐹𝑋) ⊆ ran 𝐹
15 forn 6075 . . . . . . 7 (𝐹:𝑍onto𝑌 → ran 𝐹 = 𝑌)
16153ad2ant2 1081 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ran 𝐹 = 𝑌)
1714, 16syl5sseq 3632 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) ⊆ 𝑌)
18 foima 6077 . . . . . . 7 (𝐹:𝑍onto𝑌 → (𝐹𝑍) = 𝑌)
19183ad2ant2 1081 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) = 𝑌)
20 imass2 5460 . . . . . . 7 (𝑍𝑋 → (𝐹𝑍) ⊆ (𝐹𝑋))
218, 20syl 17 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) ⊆ (𝐹𝑋))
2219, 21eqsstr3d 3619 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑌 ⊆ (𝐹𝑋))
2317, 22eqssd 3600 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) = 𝑌)
2423pweqd 4135 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝒫 (𝐹𝑋) = 𝒫 𝑌)
25 cnvimass 5444 . . . . . . 7 (𝐹𝑠) ⊆ dom 𝐹
26 fdm 6008 . . . . . . . 8 (𝐹:𝑍𝑌 → dom 𝐹 = 𝑍)
273, 26syl 17 . . . . . . 7 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → dom 𝐹 = 𝑍)
2825, 27syl5sseq 3632 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑍)
2928, 8sstrd 3593 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑋)
30 df-ss 3569 . . . . 5 ((𝐹𝑠) ⊆ 𝑋 ↔ ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
3129, 30sylib 208 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
3231eleq1d 2683 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (((𝐹𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (𝐹𝑠) ∈ 𝐽))
3324, 32rabeqbidv 3181 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
3413, 33eqtrd 2655 1 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  wf 5843  ontowfo 5845  (class class class)co 6604   qTop cqtop 16084
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-qtop 16088
This theorem is referenced by:  elqtop  21410
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