Theorem qtopkgen 23675

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 23507	. . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
2		qtopcmp.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	2	qtoptop 23665	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
4	1, 3	sylan 581	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
5		elssuni 4881	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹)) → 𝑥 ⊆ ∪ (𝑘Gen‘(𝐽 qTop 𝐹)))
6	5	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ∪ (𝑘Gen‘(𝐽 qTop 𝐹)))
7	4	adantr 480	. . . . . . . 8 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ Top)
8		eqid 2736	. . . . . . . . 9 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
9	8	kgenuni 23504	. . . . . . . 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 3959	. . . . . 6 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ∪ (𝐽 qTop 𝐹))
12		simpll 767	. . . . . . . 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 769	. . . . . . . 8 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 Fn 𝑋)
15		dffn4 6758	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
16	14, 15	sylib 218	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹:𝑋–onto→ran 𝐹)
17	2	qtopuni 23667	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
18	13, 16, 17	syl2anc 585	. . . . . 6 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
19	11, 18	sseqtrrd 3959	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ran 𝐹)
20	2	toptopon 22882	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
21	13, 20	sylib 218	. . . . . . . 8 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐽 ∈ (TopOn‘𝑋))
22		qtopid 23670	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
23	21, 14, 22	syl2anc 585	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
24		kgencn3 23523	. . . . . . . 8 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ (𝐽 qTop 𝐹) ∈ Top) → (𝐽 Cn (𝐽 qTop 𝐹)) = (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
25	12, 7, 24	syl2anc 585	. . . . . . 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 2838	. . . . . 6 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
27		cnima 23230	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (◡𝐹 “ 𝑥) ∈ 𝐽)
28	26, 27	sylancom 589	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (◡𝐹 “ 𝑥) ∈ 𝐽)
29	2	elqtop2 23666	. . . . . 6 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
30	12, 16, 29	syl2anc 585	. . . . 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 714	. . . 4 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ∈ (𝐽 qTop 𝐹))
32	31	ex 412	. . 3 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹)) → 𝑥 ∈ (𝐽 qTop 𝐹)))
33	32	ssrdv 3927	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹))
34		iskgen2 23513	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ ran 𝑘Gen ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹)))
35	4, 33, 34	sylanbrc 584	1 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ ran 𝑘Gen)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  ∪ cuni 4850  ^◡ccnv 5630  ran crn 5632   “ cima 5634   Fn wfn 6493  –onto→wfo 6496  ‘cfv 6498  (class class class)co 7367   qTop cqtop 17467  Topctop 22858  TopOnctopon 22875   Cn ccn 23189  𝑘Genckgen 23498

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-1o 8405  df-map 8775  df-en 8894  df-dom 8895  df-fin 8897  df-fi 9324  df-rest 17385  df-topgen 17406  df-qtop 17471  df-top 22859  df-topon 22876  df-bases 22911  df-cn 23192  df-cmp 23352  df-kgen 23499

This theorem is referenced by: (None)