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Theorem qtopomap 21515
Description: If 𝐹 is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open 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)
qtopomap.7 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
Assertion
Ref Expression
qtopomap (𝜑𝐾 = (𝐽 qTop 𝐹))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥   𝑥,𝑌

Proof of Theorem qtopomap
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 21512 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
51, 2, 3, 4syl3anc 1325 . 2 (𝜑𝐾 ⊆ (𝐽 qTop 𝐹))
6 cntop1 21038 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
71, 6syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
8 eqid 2621 . . . . . . 7 𝐽 = 𝐽
98toptopon 20716 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
107, 9sylib 208 . . . . 5 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
11 cnf2 21047 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
1210, 2, 1, 11syl3anc 1325 . . . . . . 7 (𝜑𝐹: 𝐽𝑌)
13 ffn 6043 . . . . . . 7 (𝐹: 𝐽𝑌𝐹 Fn 𝐽)
1412, 13syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐽)
15 df-fo 5892 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1614, 3, 15sylanbrc 698 . . . . 5 (𝜑𝐹: 𝐽onto𝑌)
17 elqtop3 21500 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹: 𝐽onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
1810, 16, 17syl2anc 693 . . . 4 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
19 foimacnv 6152 . . . . . . . 8 ((𝐹: 𝐽onto𝑌𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2016, 19sylan 488 . . . . . . 7 ((𝜑𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2120adantrr 753 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
22 simprr 796 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑦) ∈ 𝐽)
23 qtopomap.7 . . . . . . . . 9 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
2423ralrimiva 2965 . . . . . . . 8 (𝜑 → ∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾)
2524adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾)
26 imaeq2 5460 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
2726eleq1d 2685 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ((𝐹𝑥) ∈ 𝐾 ↔ (𝐹 “ (𝐹𝑦)) ∈ 𝐾))
2827rspcv 3303 . . . . . . 7 ((𝐹𝑦) ∈ 𝐽 → (∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾 → (𝐹 “ (𝐹𝑦)) ∈ 𝐾))
2922, 25, 28sylc 65 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹𝑦)) ∈ 𝐾)
3021, 29eqeltrrd 2701 . . . . 5 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝐾)
3130ex 450 . . . 4 (𝜑 → ((𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽) → 𝑦𝐾))
3218, 31sylbid 230 . . 3 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦𝐾))
3332ssrdv 3607 . 2 (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
345, 33eqssd 3618 1 (𝜑𝐾 = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  wss 3572   cuni 4434  ccnv 5111  ran crn 5113  cima 5115   Fn wfn 5881  wf 5882  ontowfo 5884  cfv 5886  (class class class)co 6647   qTop cqtop 16157  Topctop 20692  TopOnctopon 20709   Cn ccn 21022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-map 7856  df-qtop 16161  df-top 20693  df-topon 20710  df-cn 21025
This theorem is referenced by:  hmeoqtop  21572
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