Theorem qtopomap 24070

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 24067	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
5	1, 2, 3, 4	syl3anc 1399	. 2 ⊢ (𝜑 → 𝐾 ⊆ (𝐽 qTop 𝐹))
6		cntop1 23591	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
7	1, 6	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
8		toptopon2 23269	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
9	7, 8	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
10		cnf2 23600	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
11	9, 2, 1, 10	syl3anc 1399	. . . . . . 7 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌)
12	11	ffnd 6735	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
13		df-fo 6567	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶onto𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌))
14	12, 3, 13	sylanbrc 595	. . . . 5 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶onto𝑌)
15		elqtop3 24055	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶onto𝑌) → (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 6867	. . . . . . . 8 ⊢ ((𝐹:∪ 𝐽⟶onto𝑌 ∧ 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 6061	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)))
21	20	eleq1d 2849	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐹 “ 𝑥) ∈ 𝐾 ↔ (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾))
22		qtopomap.7	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)
23	22	ralrimiva 3158	. . . . . . . 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 3581	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾)
27	19, 26	eqeltrrd 2865	. . . . 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 3919	. 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
31	5, 30	eqssd 3930	1 ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  ∪ cuni 4870  ^◡ccnv 5655  ran crn 5657   “ cima 5659   Fn wfn 6556  ⟶wf 6557  ⟶ontowfo 6559  ‘cfv 6561  (class class class)co 7446   qTop cqtop 17707  Topctop 23244  TopOnctopon 23261   Cn ccn 23575

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-map 8880  df-qtop 17711  df-top 23245  df-topon 23262  df-cn 23578

This theorem is used by:  hmeoqtop  24127