Theorem qtopcn 22565

Description: Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)

Assertion

Ref	Expression
qtopcn	⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))

Proof of Theorem qtopcn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 5934	. . . . . . 7 ⊢ (◡𝐺 “ 𝑥) ⊆ dom 𝐺
2		simplrr 778	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐺:𝑌⟶𝑍)
3	1, 2	fssdm 6543	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → (◡𝐺 “ 𝑥) ⊆ 𝑌)
4		simplll 775	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5		simplrl 777	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐹:𝑋–onto→𝑌)
6		elqtop3 22554	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((◡𝐺 “ 𝑥) ⊆ 𝑌 ∧ (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽)))
7	4, 5, 6	syl2anc 587	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → ((◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((◡𝐺 “ 𝑥) ⊆ 𝑌 ∧ (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽)))
8	3, 7	mpbirand 707	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → ((◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽))
9		cnvco 5739	. . . . . . . 8 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
10	9	imaeq1i 5911	. . . . . . 7 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥)
11		imaco 6095	. . . . . . 7 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
12	10, 11	eqtri 2759	. . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
13	12	eleq1i 2821	. . . . 5 ⊢ ((◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽)
14	8, 13	bitr4di 292	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → ((◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽))
15	14	ralbidva 3107	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (∀𝑥 ∈ 𝐾 (◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ∀𝑥 ∈ 𝐾 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽))
16		simprr 773	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐺:𝑌⟶𝑍)
17	16	biantrurd 536	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (∀𝑥 ∈ 𝐾 (◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐺:𝑌⟶𝑍 ∧ ∀𝑥 ∈ 𝐾 (◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹))))
18		fof 6611	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
19	18	ad2antrl 728	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐹:𝑋⟶𝑌)
20		fco 6547	. . . . 5 ⊢ ((𝐺:𝑌⟶𝑍 ∧ 𝐹:𝑋⟶𝑌) → (𝐺 ∘ 𝐹):𝑋⟶𝑍)
21	16, 19, 20	syl2anc 587	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐺 ∘ 𝐹):𝑋⟶𝑍)
22	21	biantrurd 536	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (∀𝑥 ∈ 𝐾 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽 ↔ ((𝐺 ∘ 𝐹):𝑋⟶𝑍 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)))
23	15, 17, 22	3bitr3d 312	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → ((𝐺:𝑌⟶𝑍 ∧ ∀𝑥 ∈ 𝐾 (◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹)) ↔ ((𝐺 ∘ 𝐹):𝑋⟶𝑍 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)))
24		qtoptopon 22555	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
25	24	ad2ant2r 747	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
26		simplr 769	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐾 ∈ (TopOn‘𝑍))
27		iscn 22086	. . 3 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺:𝑌⟶𝑍 ∧ ∀𝑥 ∈ 𝐾 (◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹))))
28	25, 26, 27	syl2anc 587	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺:𝑌⟶𝑍 ∧ ∀𝑥 ∈ 𝐾 (◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹))))
29		iscn 22086	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) → ((𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺 ∘ 𝐹):𝑋⟶𝑍 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)))
30	29	adantr 484	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → ((𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺 ∘ 𝐹):𝑋⟶𝑍 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)))
31	23, 28, 30	3bitr4d 314	1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  ∀wral 3051   ⊆ wss 3853  ^◡ccnv 5535   “ cima 5539   ∘ ccom 5540  ⟶wf 6354  –onto→wfo 6356  ‘cfv 6358  (class class class)co 7191   qTop cqtop 16962  TopOnctopon 21761   Cn ccn 22075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-map 8488  df-qtop 16966  df-top 21745  df-topon 21762  df-cn 22078

This theorem is referenced by:  qtopeu  22567