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Theorem qtopval2 21547
 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 1081 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽𝑉)
2 fof 6153 . . . . 5 (𝐹:𝑍onto𝑌𝐹:𝑍𝑌)
323ad2ant2 1103 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹:𝑍𝑌)
4 qtopval.1 . . . . . 6 𝑋 = 𝐽
5 uniexg 6997 . . . . . . 7 (𝐽𝑉 𝐽 ∈ V)
653ad2ant1 1102 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽 ∈ V)
74, 6syl5eqel 2734 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑋 ∈ V)
8 simp3 1083 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍𝑋)
97, 8ssexd 4838 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍 ∈ V)
10 fex 6530 . . . 4 ((𝐹:𝑍𝑌𝑍 ∈ V) → 𝐹 ∈ V)
113, 9, 10syl2anc 694 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹 ∈ V)
124qtopval 21546 . . 3 ((𝐽𝑉𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
131, 11, 12syl2anc 694 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
14 imassrn 5512 . . . . . 6 (𝐹𝑋) ⊆ ran 𝐹
15 forn 6156 . . . . . . 7 (𝐹:𝑍onto𝑌 → ran 𝐹 = 𝑌)
16153ad2ant2 1103 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ran 𝐹 = 𝑌)
1714, 16syl5sseq 3686 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) ⊆ 𝑌)
18 foima 6158 . . . . . . 7 (𝐹:𝑍onto𝑌 → (𝐹𝑍) = 𝑌)
19183ad2ant2 1103 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) = 𝑌)
20 imass2 5536 . . . . . . 7 (𝑍𝑋 → (𝐹𝑍) ⊆ (𝐹𝑋))
218, 20syl 17 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) ⊆ (𝐹𝑋))
2219, 21eqsstr3d 3673 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑌 ⊆ (𝐹𝑋))
2317, 22eqssd 3653 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) = 𝑌)
2423pweqd 4196 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝒫 (𝐹𝑋) = 𝒫 𝑌)
25 cnvimass 5520 . . . . . . 7 (𝐹𝑠) ⊆ dom 𝐹
26 fdm 6089 . . . . . . . 8 (𝐹:𝑍𝑌 → dom 𝐹 = 𝑍)
273, 26syl 17 . . . . . . 7 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → dom 𝐹 = 𝑍)
2825, 27syl5sseq 3686 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑍)
2928, 8sstrd 3646 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑋)
30 df-ss 3621 . . . . 5 ((𝐹𝑠) ⊆ 𝑋 ↔ ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
3129, 30sylib 208 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
3231eleq1d 2715 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (((𝐹𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (𝐹𝑠) ∈ 𝐽))
3324, 32rabeqbidv 3226 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
3413, 33eqtrd 2685 1 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {crab 2945  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  𝒫 cpw 4191  ∪ cuni 4468  ◡ccnv 5142  dom cdm 5143  ran crn 5144   “ cima 5146  ⟶wf 5922  –onto→wfo 5924  (class class class)co 6690   qTop cqtop 16210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-qtop 16214 This theorem is referenced by:  elqtop  21548
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