Theorem qtopid 22962

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 485	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 Fn 𝑋)
2		dffn4 6745	. . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
3	1, 2	sylib 217	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋–onto→ran 𝐹)
4		fof 6739	. . 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 22960	. . . . 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 500	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝐽 qTop 𝐹)) → (◡𝐹 “ 𝑥) ∈ 𝐽)
9	8	ralrimiva 3139	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → ∀𝑥 ∈ (𝐽 qTop 𝐹)(◡𝐹 “ 𝑥) ∈ 𝐽)
10		qtoptopon 22961	. . . 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 22492	. . 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 710	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2105  ∀wral 3061   ⊆ wss 3898  ^◡ccnv 5619  ran crn 5621   “ cima 5623   Fn wfn 6474  ⟶wf 6475  –onto→wfo 6477  ‘cfv 6479  (class class class)co 7337   qTop cqtop 17311  TopOnctopon 22165   Cn ccn 22481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-map 8688  df-qtop 17315  df-top 22149  df-topon 22166  df-cn 22484

This theorem is referenced by:  qtopcmplem  22964  qtopkgen  22967  qtoprest  22974  kqid  22985  qtopf1  23073  qtophmeo  23074  qustgplem  23378  circcn  32086