Theorem qtopid 23200

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 6808	. . . 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 6802	. . 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 23198	. . . . 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 3146	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → ∀𝑥 ∈ (𝐽 qTop 𝐹)(◡𝐹 “ 𝑥) ∈ 𝐽)
10		qtoptopon 23199	. . . 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 22730	. . 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 711	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3947  ^◡ccnv 5674  ran crn 5676   “ cima 5678   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  ‘cfv 6540  (class class class)co 7405   qTop cqtop 17445  TopOnctopon 22403   Cn ccn 22719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-qtop 17449  df-top 22387  df-topon 22404  df-cn 22722

This theorem is referenced by:  qtopcmplem  23202  qtopkgen  23205  qtoprest  23212  kqid  23223  qtopf1  23311  qtophmeo  23312  qustgplem  23616  circcn  32806