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Theorem qtopcld 21326
 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 21317 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2 topontop 20541 . . 3 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3 eqid 2610 . . . 4 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
43iscld 20641 . . 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 20542 . . . . 5 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = (𝐽 qTop 𝐹))
71, 6syl 17 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
87sseq2d 3596 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴𝑌𝐴 (𝐽 qTop 𝐹)))
97difeq1d 3689 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝑌𝐴) = ( (𝐽 qTop 𝐹) ∖ 𝐴))
109eleq1d 2672 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹)))
118, 10anbi12d 743 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐴𝑌 ∧ (𝑌𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 (𝐽 qTop 𝐹) ∧ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
12 elqtop3 21316 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽)))
1312adantr 480 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽)))
14 difss 3699 . . . . . 6 (𝑌𝐴) ⊆ 𝑌
1514biantrur 526 . . . . 5 ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽))
16 fofun 6029 . . . . . . . . . 10 (𝐹:𝑋onto𝑌 → Fun 𝐹)
1716ad2antlr 759 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → Fun 𝐹)
18 funcnvcnv 5870 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
19 imadif 5887 . . . . . . . . 9 (Fun 𝐹 → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
2017, 18, 193syl 18 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
21 fof 6028 . . . . . . . . . . . 12 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
22 fimacnv 6255 . . . . . . . . . . . 12 (𝐹:𝑋𝑌 → (𝐹𝑌) = 𝑋)
2321, 22syl 17 . . . . . . . . . . 11 (𝐹:𝑋onto𝑌 → (𝐹𝑌) = 𝑋)
2423ad2antlr 759 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝑌) = 𝑋)
25 toponuni 20542 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2625ad2antrr 758 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝑋 = 𝐽)
2724, 26eqtrd 2644 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝑌) = 𝐽)
2827difeq1d 3689 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹𝑌) ∖ (𝐹𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
2920, 28eqtrd 2644 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
3029eleq1d 2672 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
31 topontop 20541 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3231ad2antrr 758 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝐽 ∈ Top)
33 cnvimass 5404 . . . . . . . . 9 (𝐹𝐴) ⊆ dom 𝐹
34 fofn 6030 . . . . . . . . . . 11 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
35 fndm 5904 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
3634, 35syl 17 . . . . . . . . . 10 (𝐹:𝑋onto𝑌 → dom 𝐹 = 𝑋)
3736ad2antlr 759 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → dom 𝐹 = 𝑋)
3833, 37syl5sseq 3616 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝑋)
3938, 26sseqtrd 3604 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝐽)
40 eqid 2610 . . . . . . . 8 𝐽 = 𝐽
4140iscld2 20642 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝐹𝐴) ⊆ 𝐽) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
4232, 39, 41syl2anc 691 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
4330, 42bitr4d 270 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4415, 43syl5bbr 273 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4513, 44bitrd 267 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4645pm5.32da 671 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐴𝑌 ∧ (𝑌𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ (Clsd‘𝐽))))
475, 11, 463bitr2d 295 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  ∪ cuni 4372  ◡ccnv 5037  dom cdm 5038   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   qTop cqtop 15986  Topctop 20517  TopOnctopon 20518  Clsdccld 20630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-qtop 15990  df-top 20521  df-topon 20523  df-cld 20633 This theorem is referenced by:  qtoprest  21330  kqcld  21348  qustgphaus  21736  qtopt1  29230
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