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Theorem qtopcmap 21570
Description: If 𝐹 is a surjective continuous closed map, then it is a quotient map. (A closed map is a function that maps closed sets to closed sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
qtopomap.4 (𝜑𝐾 ∈ (TopOn‘𝑌))
qtopomap.5 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
qtopomap.6 (𝜑 → ran 𝐹 = 𝑌)
qtopcmap.7 ((𝜑𝑥 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
Assertion
Ref Expression
qtopcmap (𝜑𝐾 = (𝐽 qTop 𝐹))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥   𝑥,𝑌

Proof of Theorem qtopcmap
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 qtopomap.5 . . 3 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
2 qtopomap.4 . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 qtopomap.6 . . 3 (𝜑 → ran 𝐹 = 𝑌)
4 qtopss 21566 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
51, 2, 3, 4syl3anc 1366 . 2 (𝜑𝐾 ⊆ (𝐽 qTop 𝐹))
6 cntop1 21092 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
71, 6syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
8 eqid 2651 . . . . . . . . . 10 𝐽 = 𝐽
98toptopon 20770 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
107, 9sylib 208 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
11 cnf2 21101 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
1210, 2, 1, 11syl3anc 1366 . . . . . . 7 (𝜑𝐹: 𝐽𝑌)
13 ffn 6083 . . . . . . 7 (𝐹: 𝐽𝑌𝐹 Fn 𝐽)
1412, 13syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐽)
15 df-fo 5932 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1614, 3, 15sylanbrc 699 . . . . 5 (𝜑𝐹: 𝐽onto𝑌)
178elqtop2 21552 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹: 𝐽onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
187, 16, 17syl2anc 694 . . . 4 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
1916adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽onto𝑌)
20 difss 3770 . . . . . . . . 9 (𝑌𝑦) ⊆ 𝑌
21 foimacnv 6192 . . . . . . . . 9 ((𝐹: 𝐽onto𝑌 ∧ (𝑌𝑦) ⊆ 𝑌) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
2219, 20, 21sylancl 695 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
232adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ (TopOn‘𝑌))
24 toponuni 20767 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2523, 24syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑌 = 𝐾)
2625difeq1d 3760 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑌𝑦) = ( 𝐾𝑦))
2722, 26eqtrd 2685 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = ( 𝐾𝑦))
28 fofun 6154 . . . . . . . . . . 11 (𝐹: 𝐽onto𝑌 → Fun 𝐹)
29 funcnvcnv 5994 . . . . . . . . . . 11 (Fun 𝐹 → Fun 𝐹)
30 imadif 6011 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3119, 28, 29, 304syl 19 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3212adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽𝑌)
33 fimacnv 6387 . . . . . . . . . . . 12 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
3432, 33syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑌) = 𝐽)
3534difeq1d 3760 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ((𝐹𝑌) ∖ (𝐹𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
3631, 35eqtrd 2685 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
377adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐽 ∈ Top)
38 simprr 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑦) ∈ 𝐽)
398opncld 20885 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑦) ∈ 𝐽) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4037, 38, 39syl2anc 694 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4136, 40eqeltrd 2730 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) ∈ (Clsd‘𝐽))
42 qtopcmap.7 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
4342ralrimiva 2995 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
4443adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
45 imaeq2 5497 . . . . . . . . . 10 (𝑥 = (𝐹 “ (𝑌𝑦)) → (𝐹𝑥) = (𝐹 “ (𝐹 “ (𝑌𝑦))))
4645eleq1d 2715 . . . . . . . . 9 (𝑥 = (𝐹 “ (𝑌𝑦)) → ((𝐹𝑥) ∈ (Clsd‘𝐾) ↔ (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾)))
4746rspcv 3336 . . . . . . . 8 ((𝐹 “ (𝑌𝑦)) ∈ (Clsd‘𝐽) → (∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾) → (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾)))
4841, 44, 47sylc 65 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾))
4927, 48eqeltrrd 2731 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐾𝑦) ∈ (Clsd‘𝐾))
50 topontop 20766 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
5123, 50syl 17 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ Top)
52 simprl 809 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝑌)
5352, 25sseqtrd 3674 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦 𝐾)
54 eqid 2651 . . . . . . . 8 𝐾 = 𝐾
5554isopn2 20884 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑦 𝐾) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5651, 53, 55syl2anc 694 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5749, 56mpbird 247 . . . . 5 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝐾)
5857ex 449 . . . 4 (𝜑 → ((𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽) → 𝑦𝐾))
5918, 58sylbid 230 . . 3 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦𝐾))
6059ssrdv 3642 . 2 (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
615, 60eqssd 3653 1 (𝜑𝐾 = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  cdif 3604  wss 3607   cuni 4468  ccnv 5142  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690   qTop cqtop 16210  Topctop 20746  TopOnctopon 20763  Clsdccld 20868   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-qtop 16214  df-top 20747  df-topon 20764  df-cld 20871  df-cn 21079
This theorem is referenced by: (None)
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