Theorem qtopid 23640

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 6749	. . . 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 6743	. . 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 23638	. . . . 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 3125	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → ∀𝑥 ∈ (𝐽 qTop 𝐹)(◡𝐹 “ 𝑥) ∈ 𝐽)
10		qtoptopon 23639	. . . 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 23170	. . 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 2113  ∀wral 3048   ⊆ wss 3898  ^◡ccnv 5620  ran crn 5622   “ cima 5624   Fn wfn 6484  ⟶wf 6485  –onto→wfo 6487  ‘cfv 6489  (class class class)co 7355   qTop cqtop 17415  TopOnctopon 22845   Cn ccn 23159

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-map 8761  df-qtop 17419  df-top 22829  df-topon 22846  df-cn 23162

This theorem is referenced by:  qtopcmplem  23642  qtopkgen  23645  qtoprest  23652  kqid  23663  qtopf1  23751  qtophmeo  23752  qustgplem  24056  circcn  33923