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Theorem qtopcld 23208
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 23199 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
2 topontop 22406 . . 3 ((𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ) β†’ (𝐽 qTop 𝐹) ∈ Top)
3 eqid 2732 . . . 4 βˆͺ (𝐽 qTop 𝐹) = βˆͺ (𝐽 qTop 𝐹)
43iscld 22522 . . 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 22407 . . . . 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 4013 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐴 βŠ† π‘Œ ↔ 𝐴 βŠ† βˆͺ (𝐽 qTop 𝐹)))
97difeq1d 4120 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (π‘Œ βˆ– 𝐴) = (βˆͺ (𝐽 qTop 𝐹) βˆ– 𝐴))
109eleq1d 2818 . . 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 631 . 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 23198 . . . . 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 481 . . . 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 4130 . . . . . 6 (π‘Œ βˆ– 𝐴) βŠ† π‘Œ
1514biantrur 531 . . . . 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 6803 . . . . . . . . . 10 (𝐹:𝑋–ontoβ†’π‘Œ β†’ Fun 𝐹)
1716ad2antlr 725 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ Fun 𝐹)
18 funcnvcnv 6612 . . . . . . . . 9 (Fun 𝐹 β†’ Fun ◑◑𝐹)
19 imadif 6629 . . . . . . . . 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 6802 . . . . . . . . . . . 12 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
22 fimacnv 6736 . . . . . . . . . . . 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 725 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ π‘Œ) = 𝑋)
25 toponuni 22407 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2625ad2antrr 724 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ 𝑋 = βˆͺ 𝐽)
2724, 26eqtrd 2772 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ π‘Œ) = βˆͺ 𝐽)
2827difeq1d 4120 . . . . . . . 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 2772 . . . . . . 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 2818 . . . . . 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 22406 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
3231ad2antrr 724 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ 𝐽 ∈ Top)
33 cnvimass 6077 . . . . . . . . 9 (◑𝐹 β€œ 𝐴) βŠ† dom 𝐹
34 fofn 6804 . . . . . . . . . . 11 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹 Fn 𝑋)
3534fndmd 6651 . . . . . . . . . 10 (𝐹:𝑋–ontoβ†’π‘Œ β†’ dom 𝐹 = 𝑋)
3635ad2antlr 725 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ dom 𝐹 = 𝑋)
3733, 36sseqtrid 4033 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ 𝐴) βŠ† 𝑋)
3837, 26sseqtrd 4021 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ 𝐴) βŠ† βˆͺ 𝐽)
39 eqid 2732 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
4039iscld2 22523 . . . . . . 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 584 . . . . . 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 281 . . . . 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 284 . . . 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 278 . . 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 579 . 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 306 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 396   = wceq 1541   ∈ wcel 2106   βˆ– cdif 3944   βŠ† wss 3947  βˆͺ cuni 4907  β—‘ccnv 5674  dom cdm 5675   β€œ cima 5678  Fun wfun 6534  βŸΆwf 6536  β€“ontoβ†’wfo 6538  β€˜cfv 6540  (class class class)co 7405   qTop cqtop 17445  Topctop 22386  TopOnctopon 22403  Clsdccld 22511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-qtop 17449  df-top 22387  df-topon 22404  df-cld 22514
This theorem is referenced by:  qtoprest  23212  kqcld  23230  qustgphaus  23618  qtopt1  32803
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