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Theorem qtopval2 21988
 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 1129 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽𝑉)
2 fof 6458 . . . . 5 (𝐹:𝑍onto𝑌𝐹:𝑍𝑌)
323ad2ant2 1127 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹:𝑍𝑌)
4 qtopval.1 . . . . . 6 𝑋 = 𝐽
5 uniexg 7325 . . . . . . 7 (𝐽𝑉 𝐽 ∈ V)
653ad2ant1 1126 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐽 ∈ V)
74, 6syl5eqel 2887 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑋 ∈ V)
8 simp3 1131 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍𝑋)
97, 8ssexd 5119 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑍 ∈ V)
10 fex 6855 . . . 4 ((𝐹:𝑍𝑌𝑍 ∈ V) → 𝐹 ∈ V)
113, 9, 10syl2anc 584 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝐹 ∈ V)
124qtopval 21987 . . 3 ((𝐽𝑉𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
131, 11, 12syl2anc 584 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
14 imassrn 5817 . . . . . 6 (𝐹𝑋) ⊆ ran 𝐹
15 forn 6461 . . . . . . 7 (𝐹:𝑍onto𝑌 → ran 𝐹 = 𝑌)
16153ad2ant2 1127 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ran 𝐹 = 𝑌)
1714, 16sseqtrid 3940 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) ⊆ 𝑌)
18 foima 6463 . . . . . . 7 (𝐹:𝑍onto𝑌 → (𝐹𝑍) = 𝑌)
19183ad2ant2 1127 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) = 𝑌)
20 imass2 5841 . . . . . . 7 (𝑍𝑋 → (𝐹𝑍) ⊆ (𝐹𝑋))
218, 20syl 17 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑍) ⊆ (𝐹𝑋))
2219, 21eqsstrrd 3927 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝑌 ⊆ (𝐹𝑋))
2317, 22eqssd 3906 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑋) = 𝑌)
2423pweqd 4458 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → 𝒫 (𝐹𝑋) = 𝒫 𝑌)
25 cnvimass 5825 . . . . . . 7 (𝐹𝑠) ⊆ dom 𝐹
2625, 3fssdm 6398 . . . . . 6 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑍)
2726, 8sstrd 3899 . . . . 5 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐹𝑠) ⊆ 𝑋)
28 df-ss 3874 . . . . 5 ((𝐹𝑠) ⊆ 𝑋 ↔ ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
2927, 28sylib 219 . . . 4 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → ((𝐹𝑠) ∩ 𝑋) = (𝐹𝑠))
3029eleq1d 2867 . . 3 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (((𝐹𝑠) ∩ 𝑋) ∈ 𝐽 ↔ (𝐹𝑠) ∈ 𝐽))
3124, 30rabeqbidv 3430 . 2 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽} = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
3213, 31eqtrd 2831 1 ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  {crab 3109  Vcvv 3437   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4453  ∪ cuni 4745  ◡ccnv 5442  ran crn 5444   “ cima 5446  ⟶wf 6221  –onto→wfo 6223  (class class class)co 7016   qTop cqtop 16605 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-qtop 16609 This theorem is referenced by:  elqtop  21989
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