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Theorem qtopcld 21456
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 21447 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2 topontop 20658 . . 3 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3 eqid 2621 . . . 4 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
43iscld 20771 . . 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 20659 . . . . 5 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = (𝐽 qTop 𝐹))
71, 6syl 17 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
87sseq2d 3618 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴𝑌𝐴 (𝐽 qTop 𝐹)))
97difeq1d 3711 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝑌𝐴) = ( (𝐽 qTop 𝐹) ∖ 𝐴))
109eleq1d 2683 . . 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 21446 . . . . 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 3721 . . . . . 6 (𝑌𝐴) ⊆ 𝑌
1514biantrur 527 . . . . 5 ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽))
16 fofun 6083 . . . . . . . . . 10 (𝐹:𝑋onto𝑌 → Fun 𝐹)
1716ad2antlr 762 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → Fun 𝐹)
18 funcnvcnv 5924 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
19 imadif 5941 . . . . . . . . 9 (Fun 𝐹 → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
2017, 18, 193syl 18 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
21 fof 6082 . . . . . . . . . . . 12 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
22 fimacnv 6313 . . . . . . . . . . . 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 20659 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2625ad2antrr 761 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝑋 = 𝐽)
2724, 26eqtrd 2655 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝑌) = 𝐽)
2827difeq1d 3711 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹𝑌) ∖ (𝐹𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
2920, 28eqtrd 2655 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
3029eleq1d 2683 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
31 topontop 20658 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3231ad2antrr 761 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝐽 ∈ Top)
33 cnvimass 5454 . . . . . . . . 9 (𝐹𝐴) ⊆ dom 𝐹
34 fofn 6084 . . . . . . . . . . 11 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
35 fndm 5958 . . . . . . . . . . 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 3638 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝑋)
3938, 26sseqtrd 3626 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝐽)
40 eqid 2621 . . . . . . . 8 𝐽 = 𝐽
4140iscld2 20772 . . . . . . 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 1987  cdif 3557  wss 3560   cuni 4409  ccnv 5083  dom cdm 5084  cima 5087  Fun wfun 5851   Fn wfn 5852  wf 5853  ontowfo 5855  cfv 5857  (class class class)co 6615   qTop cqtop 16103  Topctop 20638  TopOnctopon 20655  Clsdccld 20760
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-qtop 16107  df-top 20639  df-topon 20656  df-cld 20763
This theorem is referenced by:  qtoprest  21460  kqcld  21478  qustgphaus  21866  qtopt1  29726
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