Theorem qtopomap 23950

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	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
qtopomap.5	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
qtopomap.6	⊢ (𝜑 → ran 𝐹 = 𝑌)
qtopomap.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)

Assertion

Ref	Expression
qtopomap	⊢ (𝜑 → 𝐾 = (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.

Step	Hyp	Ref	Expression
1		qtopomap.5	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
2		qtopomap.4	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		qtopomap.6	. . 3 ⊢ (𝜑 → ran 𝐹 = 𝑌)
4		qtopss 23947	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
5	1, 2, 3, 4	syl3anc 1398	. 2 ⊢ (𝜑 → 𝐾 ⊆ (𝐽 qTop 𝐹))
6		cntop1 23471	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
7	1, 6	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
8		toptopon2 23149	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
9	7, 8	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
10		cnf2 23480	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
11	9, 2, 1, 10	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌)
12	11	ffnd 6707	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
13		df-fo 6543	. . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌))
14	12, 3, 13	sylanbrc 595	. . . . 5 ⊢ (𝜑 → 𝐹:∪ 𝐽–onto→𝑌)
15		elqtop3 23935	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹:∪ 𝐽–onto→𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))
16	9, 14, 15	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))
17		foimacnv 6839	. . . . . . . 8 ⊢ ((𝐹:∪ 𝐽–onto→𝑌 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
18	14, 17	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
19	18	adantrr 730	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
20		imaeq2 6056	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)))
21	20	eleq1d 2847	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐹 “ 𝑥) ∈ 𝐾 ↔ (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾))
22		qtopomap.7	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)
23	22	ralrimiva 3156	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐽 (𝐹 “ 𝑥) ∈ 𝐾)
24	23	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → ∀𝑥 ∈ 𝐽 (𝐹 “ 𝑥) ∈ 𝐾)
25		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ 𝑦) ∈ 𝐽)
26	21, 24, 25	rspcdva 3580	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾)
27	19, 26	eqeltrrd 2863	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ∈ 𝐾)
28	27	ex 418	. . . 4 ⊢ (𝜑 → ((𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽) → 𝑦 ∈ 𝐾))
29	16, 28	sylbid 243	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦 ∈ 𝐾))
30	29	ssrdv 3940	. 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
31	5, 30	eqssd 3951	1 ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  ∪ cuni 4870  ^◡ccnv 5658  ran crn 5660   “ cima 5662   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7417   qTop cqtop 17595  Topctop 23124  TopOnctopon 23141   Cn ccn 23455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-map 8832  df-qtop 17599  df-top 23125  df-topon 23142  df-cn 23458

This theorem is used by:  hmeoqtop  24007