Theorem qtopcn 21517

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		simplll 798	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2		simplrl 800	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐹:𝑋–onto→𝑌)
3		elqtop3 21506	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((◡𝐺 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((◡𝐺 “ 𝑥) ⊆ 𝑌 ∧ (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽)))
4	1, 2, 3	syl2anc 693	. . . . . 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)))
5		cnvimass 5485	. . . . . . . 8 ⊢ (◡𝐺 “ 𝑥) ⊆ dom 𝐺
6		simplrr 801	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐺:𝑌⟶𝑍)
7		fdm 6051	. . . . . . . . 9 ⊢ (𝐺:𝑌⟶𝑍 → dom 𝐺 = 𝑌)
8	6, 7	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → dom 𝐺 = 𝑌)
9	5, 8	syl5sseq 3653	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → (◡𝐺 “ 𝑥) ⊆ 𝑌)
10	9	biantrurd 529	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → ((◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽 ↔ ((◡𝐺 “ 𝑥) ⊆ 𝑌 ∧ (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽)))
11	4, 10	bitr4d 271	. . . . 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))
12		cnvco 5308	. . . . . . . 8 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
13	12	imaeq1i 5463	. . . . . . 7 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥)
14		imaco 5640	. . . . . . 7 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
15	13, 14	eqtri 2644	. . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
16	15	eleq1i 2692	. . . . 5 ⊢ ((◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽)
17	11, 16	syl6bbr 278	. . . 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))
18	17	ralbidva 2985	. . 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))
19		simprr 796	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐺:𝑌⟶𝑍)
20	19	biantrurd 529	. . 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))))
21		fof 6115	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
22	21	ad2antrl 764	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐹:𝑋⟶𝑌)
23		fco 6058	. . . . 5 ⊢ ((𝐺:𝑌⟶𝑍 ∧ 𝐹:𝑋⟶𝑌) → (𝐺 ∘ 𝐹):𝑋⟶𝑍)
24	19, 22, 23	syl2anc 693	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐺 ∘ 𝐹):𝑋⟶𝑍)
25	24	biantrurd 529	. . 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)))
26	18, 20, 25	3bitr3d 298	. 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)))
27		qtoptopon 21507	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
28	27	ad2ant2r 783	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
29		simplr 792	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐾 ∈ (TopOn‘𝑍))
30		iscn 21039	. . 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))))
31	28, 29, 30	syl2anc 693	. 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))))
32		iscn 21039	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) → ((𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐾) ↔ ((𝐺 ∘ 𝐹):𝑋⟶𝑍 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)))
33	32	adantr 481	. 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)))
34	26, 31, 33	3bitr4d 300	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  ^◡ccnv 5113  dom cdm 5114   “ cima 5117   ∘ ccom 5118  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650   qTop cqtop 16163  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-qtop 16167  df-top 20699  df-topon 20716  df-cn 21031

This theorem is referenced by:  qtopeu  21519