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Theorem qtopcld 21796
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 21787 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2 topontop 20997 . . 3 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3 eqid 2765 . . . 4 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
43iscld 21111 . . 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 20998 . . . . 5 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = (𝐽 qTop 𝐹))
71, 6syl 17 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
87sseq2d 3793 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴𝑌𝐴 (𝐽 qTop 𝐹)))
97difeq1d 3889 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝑌𝐴) = ( (𝐽 qTop 𝐹) ∖ 𝐴))
109eleq1d 2829 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹)))
118, 10anbi12d 624 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐴𝑌 ∧ (𝑌𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 (𝐽 qTop 𝐹) ∧ ( (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
12 elqtop3 21786 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽)))
1312adantr 472 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽)))
14 difss 3899 . . . . . 6 (𝑌𝐴) ⊆ 𝑌
1514biantrur 526 . . . . 5 ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽))
16 fofun 6299 . . . . . . . . . 10 (𝐹:𝑋onto𝑌 → Fun 𝐹)
1716ad2antlr 718 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → Fun 𝐹)
18 funcnvcnv 6134 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
19 imadif 6151 . . . . . . . . 9 (Fun 𝐹 → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
2017, 18, 193syl 18 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ((𝐹𝑌) ∖ (𝐹𝐴)))
21 fof 6298 . . . . . . . . . . . 12 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
22 fimacnv 6537 . . . . . . . . . . . 12 (𝐹:𝑋𝑌 → (𝐹𝑌) = 𝑋)
2321, 22syl 17 . . . . . . . . . . 11 (𝐹:𝑋onto𝑌 → (𝐹𝑌) = 𝑋)
2423ad2antlr 718 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝑌) = 𝑋)
25 toponuni 20998 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2625ad2antrr 717 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝑋 = 𝐽)
2724, 26eqtrd 2799 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝑌) = 𝐽)
2827difeq1d 3889 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹𝑌) ∖ (𝐹𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
2920, 28eqtrd 2799 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹 “ (𝑌𝐴)) = ( 𝐽 ∖ (𝐹𝐴)))
3029eleq1d 2829 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
31 topontop 20997 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3231ad2antrr 717 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → 𝐽 ∈ Top)
33 cnvimass 5667 . . . . . . . . 9 (𝐹𝐴) ⊆ dom 𝐹
34 fofn 6300 . . . . . . . . . . 11 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
35 fndm 6168 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
3634, 35syl 17 . . . . . . . . . 10 (𝐹:𝑋onto𝑌 → dom 𝐹 = 𝑋)
3736ad2antlr 718 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → dom 𝐹 = 𝑋)
3833, 37syl5sseq 3813 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝑋)
3938, 26sseqtrd 3801 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (𝐹𝐴) ⊆ 𝐽)
40 eqid 2765 . . . . . . . 8 𝐽 = 𝐽
4140iscld2 21112 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝐹𝐴) ⊆ 𝐽) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
4232, 39, 41syl2anc 579 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹𝐴) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐹𝐴)) ∈ 𝐽))
4330, 42bitr4d 273 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝐹 “ (𝑌𝐴)) ∈ 𝐽 ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4415, 43syl5bbr 276 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → (((𝑌𝐴) ⊆ 𝑌 ∧ (𝐹 “ (𝑌𝐴)) ∈ 𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4513, 44bitrd 270 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) ∧ 𝐴𝑌) → ((𝑌𝐴) ∈ (𝐽 qTop 𝐹) ↔ (𝐹𝐴) ∈ (Clsd‘𝐽)))
4645pm5.32da 574 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ((𝐴𝑌 ∧ (𝑌𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ (Clsd‘𝐽))))
475, 11, 463bitr2d 298 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 197  wa 384   = wceq 1652  wcel 2155  cdif 3729  wss 3732   cuni 4594  ccnv 5276  dom cdm 5277  cima 5280  Fun wfun 6062   Fn wfn 6063  wf 6064  ontowfo 6066  cfv 6068  (class class class)co 6842   qTop cqtop 16429  Topctop 20977  TopOnctopon 20994  Clsdccld 21100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-qtop 16433  df-top 20978  df-topon 20995  df-cld 21103
This theorem is referenced by:  qtoprest  21800  kqcld  21818  qustgphaus  22205  qtopt1  30349
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