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Theorem qtopcld 21421
Description: The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
Assertion
Ref Expression
qtopcld ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ (Clsd‘𝐽))))

Proof of Theorem qtopcld
StepHypRef Expression
1 qtoptopon 21412 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2 topontop 20636 . . 3 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3 eqid 2626 . . . 4 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
43iscld 20736 . . 3 ((𝐽 qTop 𝐹) ∈ Top → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 (𝐽 qTop 𝐹) ∧ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
51, 2, 43syl 18 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 (𝐽 qTop 𝐹) ∧ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
6 toponuni 20637 . . . . 5 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = (𝐽 qTop 𝐹))
71, 6syl 17 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
87sseq2d 3617 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴𝑌𝐴 (𝐽 qTop 𝐹)))
97difeq1d 3710 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝑌𝐴) = ( (𝐽 qTop 𝐹) ∖ 𝐴))
109eleq1d 2688 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹)))
118, 10anbi12d 746 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐴𝑌 ∧ (𝑌𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 (𝐽 qTop 𝐹) ∧ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
12 elqtop3 21411 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽)))
1312adantr 481 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽)))
14 difss 3720 . . . . . 6 (𝑌𝐴) ⊆ 𝑌
1514biantrur 527 . . . . 5 ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽))
16 fofun 6075 . . . . . . . . . 10 (𝐹:𝑋onto𝑌 → Fun 𝐹)
1716ad2antlr 762 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → Fun 𝐹)
18 funcnvcnv 5916 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
19 imadif 5933 . . . . . . . . 9 (Fun 𝐹 → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
2017, 18, 193syl 18 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
21 fof 6074 . . . . . . . . . . . 12 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
22 fimacnv 6304 . . . . . . . . . . . 12 (𝐹:𝑋𝑌 → (𝐹𝑌) = 𝑋)
2321, 22syl 17 . . . . . . . . . . 11 (𝐹:𝑋onto𝑌 → (𝐹𝑌) = 𝑋)
2423ad2antlr 762 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝑌) = 𝑋)
25 toponuni 20637 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2625ad2antrr 761 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝑋 = 𝐽)
2724, 26eqtrd 2660 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝑌) = 𝐽)
2827difeq1d 3710 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹𝑌) ∖ (𝐹𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
2920, 28eqtrd 2660 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
3029eleq1d 2688 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
31 topontop 20636 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3231ad2antrr 761 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝐽 ∈ Top)
33 cnvimass 5448 . . . . . . . . 9 (𝐹𝐴) ⊆ dom 𝐹
34 fofn 6076 . . . . . . . . . . 11 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
35 fndm 5950 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
3634, 35syl 17 . . . . . . . . . 10 (𝐹:𝑋onto𝑌 → dom 𝐹 = 𝑋)
3736ad2antlr 762 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → dom 𝐹 = 𝑋)
3833, 37syl5sseq 3637 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝑋)
3938, 26sseqtrd 3625 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝐽)
40 eqid 2626 . . . . . . . 8 𝐽 = 𝐽
4140iscld2 20737 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝐹𝐴) ⊆ 𝐽) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
4232, 39, 41syl2anc 692 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
4330, 42bitr4d 271 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4415, 43syl5bbr 274 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4513, 44bitrd 268 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4645pm5.32da 672 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐴𝑌 ∧ (𝑌𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ (Clsd‘𝐽))))
475, 11, 463bitr2d 296 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ (Clsd‘𝐽))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  cdif 3557  wss 3560   cuni 4407  ccnv 5078  dom cdm 5079  cima 5082  Fun wfun 5844   Fn wfn 5845  wf 5846  ontowfo 5848  cfv 5850  (class class class)co 6605   qTop cqtop 16079  Topctop 20612  TopOnctopon 20613  Clsdccld 20725
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-qtop 16083  df-top 20616  df-topon 20618  df-cld 20728
This theorem is referenced by:  qtoprest  21425  kqcld  21443  qustgphaus  21831  qtopt1  29676
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