Theorem qtopcn 23218

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 6081	. . . . . . 7 ⊢ (◡𝐺 “ 𝑥) ⊆ dom 𝐺
2		simplrr 777	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐺:𝑌⟶𝑍)
3	1, 2	fssdm 6738	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → (◡𝐺 “ 𝑥) ⊆ 𝑌)
4		simplll 774	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5		simplrl 776	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐹:𝑋–onto→𝑌)
6		elqtop3 23207	. . . . . . 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 706	. . . . 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 5886	. . . . . . . 8 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
10	9	imaeq1i 6057	. . . . . . 7 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥)
11		imaco 6251	. . . . . . 7 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
12	10, 11	eqtri 2761	. . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
13	12	eleq1i 2825	. . . . 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 3176	. . 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 772	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐺:𝑌⟶𝑍)
17	16	biantrurd 534	. . 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 6806	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
19	18	ad2antrl 727	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐹:𝑋⟶𝑌)
20		fco 6742	. . . . 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 534	. . 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 23208	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
25	24	ad2ant2r 746	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
26		simplr 768	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐾 ∈ (TopOn‘𝑍))
27		iscn 22739	. . 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 22739	. . 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 482	. 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 205   ∧ wa 397   ∈ wcel 2107  ∀wral 3062   ⊆ wss 3949  ^◡ccnv 5676   “ cima 5680   ∘ ccom 5681  ⟶wf 6540  –onto→wfo 6542  ‘cfv 6544  (class class class)co 7409   qTop cqtop 17449  TopOnctopon 22412   Cn ccn 22728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-qtop 17453  df-top 22396  df-topon 22413  df-cn 22731

This theorem is referenced by:  qtopeu  23220