Theorem qtopkgen 23658

Description: A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)

Hypothesis

Ref	Expression
qtopcmp.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
qtopkgen	⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ ran 𝑘Gen)

Proof of Theorem qtopkgen

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kgentop 23490	. . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
2		qtopcmp.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	2	qtoptop 23648	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
4	1, 3	sylan 578	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
5		elssuni 4941	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹)) → 𝑥 ⊆ ∪ (𝑘Gen‘(𝐽 qTop 𝐹)))
6	5	adantl 480	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ∪ (𝑘Gen‘(𝐽 qTop 𝐹)))
7	4	adantr 479	. . . . . . . 8 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ Top)
8		eqid 2725	. . . . . . . . 9 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
9	8	kgenuni 23487	. . . . . . . 8 ⊢ ((𝐽 qTop 𝐹) ∈ Top → ∪ (𝐽 qTop 𝐹) = ∪ (𝑘Gen‘(𝐽 qTop 𝐹)))
10	7, 9	syl 17	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → ∪ (𝐽 qTop 𝐹) = ∪ (𝑘Gen‘(𝐽 qTop 𝐹)))
11	6, 10	sseqtrrd 4018	. . . . . 6 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ∪ (𝐽 qTop 𝐹))
12		simpll 765	. . . . . . . 8 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐽 ∈ ran 𝑘Gen)
13	12, 1	syl 17	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐽 ∈ Top)
14		simplr 767	. . . . . . . 8 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 Fn 𝑋)
15		dffn4 6816	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
16	14, 15	sylib 217	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹:𝑋–onto→ran 𝐹)
17	2	qtopuni 23650	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
18	13, 16, 17	syl2anc 582	. . . . . 6 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
19	11, 18	sseqtrrd 4018	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ran 𝐹)
20	2	toptopon 22863	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
21	13, 20	sylib 217	. . . . . . . 8 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐽 ∈ (TopOn‘𝑋))
22		qtopid 23653	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
23	21, 14, 22	syl2anc 582	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
24		kgencn3 23506	. . . . . . . 8 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ (𝐽 qTop 𝐹) ∈ Top) → (𝐽 Cn (𝐽 qTop 𝐹)) = (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
25	12, 7, 24	syl2anc 582	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐽 Cn (𝐽 qTop 𝐹)) = (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
26	23, 25	eleqtrd 2827	. . . . . 6 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
27		cnima 23213	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (◡𝐹 “ 𝑥) ∈ 𝐽)
28	26, 27	sylancom 586	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (◡𝐹 “ 𝑥) ∈ 𝐽)
29	2	elqtop2 23649	. . . . . 6 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
30	12, 16, 29	syl2anc 582	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
31	19, 28, 30	mpbir2and 711	. . . 4 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ∈ (𝐽 qTop 𝐹))
32	31	ex 411	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹)) → 𝑥 ∈ (𝐽 qTop 𝐹)))
33	32	ssrdv 3982	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹))
34		iskgen2 23496	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ ran 𝑘Gen ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹)))
35	4, 33, 34	sylanbrc 581	1 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ ran 𝑘Gen)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ⊆ wss 3944  ∪ cuni 4909  ^◡ccnv 5677  ran crn 5679   “ cima 5681   Fn wfn 6544  –onto→wfo 6547  ‘cfv 6549  (class class class)co 7419   qTop cqtop 17488  Topctop 22839  TopOnctopon 22856   Cn ccn 23172  𝑘Genckgen 23481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-fin 8968  df-fi 9436  df-rest 17407  df-topgen 17428  df-qtop 17492  df-top 22840  df-topon 22857  df-bases 22893  df-cn 23175  df-cmp 23335  df-kgen 23482

This theorem is referenced by: (None)