Theorem qtopid 23643

Description: A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)

Assertion

Ref	Expression
qtopid	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))

Proof of Theorem qtopid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 Fn 𝑋)
2		dffn4 6796	. . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
3	1, 2	sylib 218	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋–onto→ran 𝐹)
4		fof 6790	. . 3 ⊢ (𝐹:𝑋–onto→ran 𝐹 → 𝐹:𝑋⟶ran 𝐹)
5	3, 4	syl 17	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋⟶ran 𝐹)
6		elqtop3 23641	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
7	3, 6	syldan 591	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
8	7	simplbda 499	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝐽 qTop 𝐹)) → (◡𝐹 “ 𝑥) ∈ 𝐽)
9	8	ralrimiva 3132	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → ∀𝑥 ∈ (𝐽 qTop 𝐹)(◡𝐹 “ 𝑥) ∈ 𝐽)
10		qtoptopon 23642	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
11	3, 10	syldan 591	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
12		iscn 23173	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹)) → (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ↔ (𝐹:𝑋⟶ran 𝐹 ∧ ∀𝑥 ∈ (𝐽 qTop 𝐹)(◡𝐹 “ 𝑥) ∈ 𝐽)))
13	11, 12	syldan 591	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ↔ (𝐹:𝑋⟶ran 𝐹 ∧ ∀𝑥 ∈ (𝐽 qTop 𝐹)(◡𝐹 “ 𝑥) ∈ 𝐽)))
14	5, 9, 13	mpbir2and 713	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  ^◡ccnv 5653  ran crn 5655   “ cima 5657   Fn wfn 6526  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7405   qTop cqtop 17517  TopOnctopon 22848   Cn ccn 23162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8842  df-qtop 17521  df-top 22832  df-topon 22849  df-cn 23165

This theorem is referenced by:  qtopcmplem  23645  qtopkgen  23648  qtoprest  23655  kqid  23666  qtopf1  23754  qtophmeo  23755  qustgplem  24059  circcn  33869