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Theorem qtopcld 23080
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 23071 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
2 topontop 22278 . . 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 22394 . . 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 22279 . . . . 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 3977 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐴 βŠ† π‘Œ ↔ 𝐴 βŠ† βˆͺ (𝐽 qTop 𝐹)))
97difeq1d 4082 . . . 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 23070 . . . . 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 4092 . . . . . 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 6758 . . . . . . . . . 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 6569 . . . . . . . . 9 (Fun 𝐹 β†’ Fun ◑◑𝐹)
19 imadif 6586 . . . . . . . . 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 6757 . . . . . . . . . . . 12 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
22 fimacnv 6691 . . . . . . . . . . . 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 22279 . . . . . . . . . . 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 4082 . . . . . . . 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 22278 . . . . . . . 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 6034 . . . . . . . . 9 (◑𝐹 β€œ 𝐴) βŠ† dom 𝐹
34 fofn 6759 . . . . . . . . . . 11 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹 Fn 𝑋)
3534fndmd 6608 . . . . . . . . . 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 3997 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ (◑𝐹 β€œ 𝐴) βŠ† 𝑋)
3837, 26sseqtrd 3985 . . . . . . 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 22395 . . . . . . 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 3908   βŠ† wss 3911  βˆͺ cuni 4866  β—‘ccnv 5633  dom cdm 5634   β€œ cima 5637  Fun wfun 6491  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358   qTop cqtop 17390  Topctop 22258  TopOnctopon 22275  Clsdccld 22383
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-qtop 17394  df-top 22259  df-topon 22276  df-cld 22386
This theorem is referenced by:  qtoprest  23084  kqcld  23102  qustgphaus  23490  qtopt1  32473
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