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Theorem qtopcld 22864
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 22855 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2 topontop 22062 . . 3 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3 eqid 2738 . . . 4 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
43iscld 22178 . . 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 22063 . . . . 5 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = (𝐽 qTop 𝐹))
71, 6syl 17 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
87sseq2d 3953 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴𝑌𝐴 (𝐽 qTop 𝐹)))
97difeq1d 4056 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝑌𝐴) = ( (𝐽 qTop 𝐹) ∖ 𝐴))
109eleq1d 2823 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹)))
118, 10anbi12d 631 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐴𝑌 ∧ (𝑌𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 (𝐽 qTop 𝐹) ∧ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
12 elqtop3 22854 . . . . 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 4066 . . . . . 6 (𝑌𝐴) ⊆ 𝑌
1514biantrur 531 . . . . 5 ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽))
16 fofun 6689 . . . . . . . . . 10 (𝐹:𝑋onto𝑌 → Fun 𝐹)
1716ad2antlr 724 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → Fun 𝐹)
18 funcnvcnv 6501 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
19 imadif 6518 . . . . . . . . 9 (Fun 𝐹 → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
2017, 18, 193syl 18 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
21 fof 6688 . . . . . . . . . . . 12 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
22 fimacnv 6622 . . . . . . . . . . . 12 (𝐹:𝑋𝑌 → (𝐹𝑌) = 𝑋)
2321, 22syl 17 . . . . . . . . . . 11 (𝐹:𝑋onto𝑌 → (𝐹𝑌) = 𝑋)
2423ad2antlr 724 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝑌) = 𝑋)
25 toponuni 22063 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2625ad2antrr 723 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝑋 = 𝐽)
2724, 26eqtrd 2778 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝑌) = 𝐽)
2827difeq1d 4056 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹𝑌) ∖ (𝐹𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
2920, 28eqtrd 2778 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
3029eleq1d 2823 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
31 topontop 22062 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3231ad2antrr 723 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝐽 ∈ Top)
33 cnvimass 5989 . . . . . . . . 9 (𝐹𝐴) ⊆ dom 𝐹
34 fofn 6690 . . . . . . . . . . 11 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
3534fndmd 6538 . . . . . . . . . 10 (𝐹:𝑋onto𝑌 → dom 𝐹 = 𝑋)
3635ad2antlr 724 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → dom 𝐹 = 𝑋)
3733, 36sseqtrid 3973 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝑋)
3837, 26sseqtrd 3961 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝐽)
39 eqid 2738 . . . . . . . 8 𝐽 = 𝐽
4039iscld2 22179 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝐹𝐴) ⊆ 𝐽) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
4132, 38, 40syl2anc 584 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
4230, 41bitr4d 281 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4315, 42bitr3id 285 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4413, 43bitrd 278 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4544pm5.32da 579 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐴𝑌 ∧ (𝑌𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ (Clsd‘𝐽))))
465, 11, 453bitr2d 307 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 205  wa 396   = wceq 1539  wcel 2106  cdif 3884  wss 3887   cuni 4839  ccnv 5588  dom cdm 5589  cima 5592  Fun wfun 6427  wf 6429  ontowfo 6431  cfv 6433  (class class class)co 7275   qTop cqtop 17214  Topctop 22042  TopOnctopon 22059  Clsdccld 22167
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-qtop 17218  df-top 22043  df-topon 22060  df-cld 22170
This theorem is referenced by:  qtoprest  22868  kqcld  22886  qustgphaus  23274  qtopt1  31785
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