Theorem qtopcn 23852

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 6086	. . . . . . 7 ⊢ (◡𝐺 “ 𝑥) ⊆ dom 𝐺
2		simplrr 789	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐺:𝑌⟶𝑍)
3	1, 2	fssdm 6727	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → (◡𝐺 “ 𝑥) ⊆ 𝑌)
4		simplll 786	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5		simplrl 788	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) ∧ 𝑥 ∈ 𝐾) → 𝐹:𝑋–onto→𝑌)
6		elqtop3 23841	. . . . . . 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 595	. . . . . 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 719	. . . . 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 5877	. . . . . . . 8 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
10	9	imaeq1i 6061	. . . . . . 7 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥)
11		imaco 6254	. . . . . . 7 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
12	10, 11	eqtri 2786	. . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
13	12	eleq1i 2854	. . . . 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 3186	. . 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 784	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐺:𝑌⟶𝑍)
17	16	biantrurd 541	. . 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 6794	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
19	18	ad2antrl 740	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐹:𝑋⟶𝑌)
20		fco 6732	. . . . 5 ⊢ ((𝐺:𝑌⟶𝑍 ∧ 𝐹:𝑋⟶𝑌) → (𝐺 ∘ 𝐹):𝑋⟶𝑍)
21	16, 19, 20	syl2anc 595	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐺 ∘ 𝐹):𝑋⟶𝑍)
22	21	biantrurd 541	. . 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 23842	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
25	24	ad2ant2r 759	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
26		simplr 780	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → 𝐾 ∈ (TopOn‘𝑍))
27		iscn 23373	. . 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 595	. 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 23373	. . 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 485	. 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 400   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3906  ^◡ccnv 5662   “ cima 5666   ∘ ccom 5667  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7412   qTop cqtop 17558  TopOnctopon 23048   Cn ccn 23362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-qtop 17562  df-top 23032  df-topon 23049  df-cn 23365

This theorem is referenced by:  qtopeu  23854