Theorem qtopcn 23670

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 6049	. . . . . . 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 6689	. . . . . 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 23659	. . . . . . 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 585	. . . . . 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 708	. . . . 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 5842	. . . . . . . 8 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
10	9	imaeq1i 6024	. . . . . . 7 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥)
11		imaco 6217	. . . . . . 7 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
12	10, 11	eqtri 2760	. . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
13	12	eleq1i 2828	. . . . 5 ⊢ ((◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽)
14	8, 13	bitr4di 289	. . . 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 3159	. . 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 532	. . 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 6754	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
19	18	ad2antrl 729	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐹:𝑋⟶𝑌)
20		fco 6694	. . . . 5 ⊢ ((𝐺:𝑌⟶𝑍 ∧ 𝐹:𝑋⟶𝑌) → (𝐺 ∘ 𝐹):𝑋⟶𝑍)
21	16, 19, 20	syl2anc 585	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐺 ∘ 𝐹):𝑋⟶𝑍)
22	21	biantrurd 532	. . 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 309	. 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 23660	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
25	24	ad2ant2r 748	. . 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 23191	. . 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 585	. 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 23191	. . 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 480	. 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 311	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 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  ^◡ccnv 5631   “ cima 5635   ∘ ccom 5636  ⟶wf 6496  –onto→wfo 6498  ‘cfv 6500  (class class class)co 7368   qTop cqtop 17436  TopOnctopon 22866   Cn ccn 23180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-qtop 17440  df-top 22850  df-topon 22867  df-cn 23183

This theorem is referenced by:  qtopeu  23672