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Theorem qtopcld 23217
Description: The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
Assertion
Ref Expression
qtopcld ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐴 ∈ (Clsdβ€˜(𝐽 qTop 𝐹)) ↔ (𝐴 βŠ† π‘Œ ∧ (◑𝐹 β€œ 𝐴) ∈ (Clsdβ€˜π½))))
Proof of Theorem qtopcld
StepHypRef Expression
1 qtoptopon 23208 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
2 topontop 22415 . . 3 ((𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ) β†’ (𝐽 qTop 𝐹) ∈ Top)
3 eqid 2733 . . . 4 βˆͺ (𝐽 qTop 𝐹) = βˆͺ (𝐽 qTop 𝐹)
43iscld 22531 . . 3 ((𝐽 qTop 𝐹) ∈ Top β†’ (𝐴 ∈ (Clsdβ€˜(𝐽 qTop 𝐹)) ↔ (𝐴 βŠ† βˆͺ (𝐽 qTop 𝐹) ∧ (βˆͺ (𝐽 qTop 𝐹) βˆ– 𝐴) ∈ (𝐽 qTop 𝐹))))
51, 2, 43syl 18 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐴 ∈ (Clsdβ€˜(𝐽 qTop 𝐹)) ↔ (𝐴 βŠ† βˆͺ (𝐽 qTop 𝐹) ∧ (βˆͺ (𝐽 qTop 𝐹) βˆ– 𝐴) ∈ (𝐽 qTop 𝐹))))
6 toponuni 22416 . . . . 5 ((𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ (𝐽 qTop 𝐹))
71, 6syl 17 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ π‘Œ = βˆͺ (𝐽 qTop 𝐹))
87sseq2d 4015 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐴 βŠ† π‘Œ ↔ 𝐴 βŠ† βˆͺ (𝐽 qTop 𝐹)))
97difeq1d 4122 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (π‘Œ βˆ– 𝐴) = (βˆͺ (𝐽 qTop 𝐹) βˆ– 𝐴))
109eleq1d 2819 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ ((π‘Œ βˆ– 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (βˆͺ (𝐽 qTop 𝐹) βˆ– 𝐴) ∈ (𝐽 qTop 𝐹)))
118, 10anbi12d 632 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ ((𝐴 βŠ† π‘Œ ∧ (π‘Œ βˆ– 𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 βŠ† βˆͺ (𝐽 qTop 𝐹) ∧ (βˆͺ (𝐽 qTop 𝐹) βˆ– 𝐴) ∈ (𝐽 qTop 𝐹))))
12 elqtop3 23207 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ ((π‘Œ βˆ– 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((π‘Œ βˆ– 𝐴) βŠ† π‘Œ ∧ (◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) ∈ 𝐽)))
1312adantr 482 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ ((π‘Œ βˆ– 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((π‘Œ βˆ– 𝐴) βŠ† π‘Œ ∧ (◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) ∈ 𝐽)))
14 difss 4132 . . . . . 6 (π‘Œ βˆ– 𝐴) βŠ† π‘Œ
1514biantrur 532 . . . . 5 ((◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) ∈ 𝐽 ↔ ((π‘Œ βˆ– 𝐴) βŠ† π‘Œ ∧ (◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) ∈ 𝐽))
16 fofun 6807 . . . . . . . . . 10 (𝐹:𝑋–ontoβ†’π‘Œ β†’ Fun 𝐹)
1716ad2antlr 726 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ Fun 𝐹)
18 funcnvcnv 6616 . . . . . . . . 9 (Fun 𝐹 β†’ Fun ◑◑𝐹)
19 imadif 6633 . . . . . . . . 9 (Fun ◑◑𝐹 β†’ (◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) = ((◑𝐹 β€œ π‘Œ) βˆ– (◑𝐹 β€œ 𝐴)))
2017, 18, 193syl 18 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) = ((◑𝐹 β€œ π‘Œ) βˆ– (◑𝐹 β€œ 𝐴)))
21 fof 6806 . . . . . . . . . . . 12 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
22 fimacnv 6740 . . . . . . . . . . . 12 (𝐹:π‘‹βŸΆπ‘Œ β†’ (◑𝐹 β€œ π‘Œ) = 𝑋)
2321, 22syl 17 . . . . . . . . . . 11 (𝐹:𝑋–ontoβ†’π‘Œ β†’ (◑𝐹 β€œ π‘Œ) = 𝑋)
2423ad2antlr 726 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ π‘Œ) = 𝑋)
25 toponuni 22416 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2625ad2antrr 725 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ 𝑋 = βˆͺ 𝐽)
2724, 26eqtrd 2773 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ π‘Œ) = βˆͺ 𝐽)
2827difeq1d 4122 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ ((◑𝐹 β€œ π‘Œ) βˆ– (◑𝐹 β€œ 𝐴)) = (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ 𝐴)))
2920, 28eqtrd 2773 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) = (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ 𝐴)))
3029eleq1d 2819 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ ((◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) ∈ 𝐽 ↔ (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ 𝐴)) ∈ 𝐽))
31 topontop 22415 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
3231ad2antrr 725 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ 𝐽 ∈ Top)
33 cnvimass 6081 . . . . . . . . 9 (◑𝐹 β€œ 𝐴) βŠ† dom 𝐹
34 fofn 6808 . . . . . . . . . . 11 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹 Fn 𝑋)
3534fndmd 6655 . . . . . . . . . 10 (𝐹:𝑋–ontoβ†’π‘Œ β†’ dom 𝐹 = 𝑋)
3635ad2antlr 726 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ dom 𝐹 = 𝑋)
3733, 36sseqtrid 4035 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ 𝐴) βŠ† 𝑋)
3837, 26sseqtrd 4023 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ 𝐴) βŠ† βˆͺ 𝐽)
39 eqid 2733 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
4039iscld2 22532 . . . . . . 7 ((𝐽 ∈ Top ∧ (◑𝐹 β€œ 𝐴) βŠ† βˆͺ 𝐽) β†’ ((◑𝐹 β€œ 𝐴) ∈ (Clsdβ€˜π½) ↔ (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ 𝐴)) ∈ 𝐽))
4132, 38, 40syl2anc 585 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ ((◑𝐹 β€œ 𝐴) ∈ (Clsdβ€˜π½) ↔ (βˆͺ 𝐽 βˆ– (◑𝐹 β€œ 𝐴)) ∈ 𝐽))
4230, 41bitr4d 282 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ ((◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) ∈ 𝐽 ↔ (◑𝐹 β€œ 𝐴) ∈ (Clsdβ€˜π½)))
4315, 42bitr3id 285 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (((π‘Œ βˆ– 𝐴) βŠ† π‘Œ ∧ (◑𝐹 β€œ (π‘Œ βˆ– 𝐴)) ∈ 𝐽) ↔ (◑𝐹 β€œ 𝐴) ∈ (Clsdβ€˜π½)))
4413, 43bitrd 279 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ ((π‘Œ βˆ– 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (◑𝐹 β€œ 𝐴) ∈ (Clsdβ€˜π½)))
4544pm5.32da 580 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ ((𝐴 βŠ† π‘Œ ∧ (π‘Œ βˆ– 𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 βŠ† π‘Œ ∧ (◑𝐹 β€œ 𝐴) ∈ (Clsdβ€˜π½))))
465, 11, 453bitr2d 307 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐴 ∈ (Clsdβ€˜(𝐽 qTop 𝐹)) ↔ (𝐴 βŠ† π‘Œ ∧ (◑𝐹 β€œ 𝐴) ∈ (Clsdβ€˜π½))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3946   βŠ† wss 3949  βˆͺ cuni 4909  β—‘ccnv 5676  dom cdm 5677   β€œ cima 5680  Fun wfun 6538  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409   qTop cqtop 17449  Topctop 22395  TopOnctopon 22412  Clsdccld 22520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-qtop 17453  df-top 22396  df-topon 22413  df-cld 22523
This theorem is referenced by:  qtoprest  23221  kqcld  23239  qustgphaus  23627  qtopt1  32846
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