Theorem qtopkgen 23595

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 23427	. . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
2		qtopcmp.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	2	qtoptop 23585	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
4	1, 3	sylan 580	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
5		elssuni 4888	. . . . . . . 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 2729	. . . . . . . . 9 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
9	8	kgenuni 23424	. . . . . . . 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 3973	. . . . . 6 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ∪ (𝐽 qTop 𝐹))
12		simpll 766	. . . . . . . 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 768	. . . . . . . 8 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 Fn 𝑋)
15		dffn4 6742	. . . . . . . 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 23587	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
18	13, 16, 17	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
19	11, 18	sseqtrrd 3973	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝑥 ⊆ ran 𝐹)
20	2	toptopon 22802	. . . . . . . . 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 23590	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
23	21, 14, 22	syl2anc 584	. . . . . . 7 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
24		kgencn3 23443	. . . . . . . 8 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ (𝐽 qTop 𝐹) ∈ Top) → (𝐽 Cn (𝐽 qTop 𝐹)) = (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
25	12, 7, 24	syl2anc 584	. . . . . . 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 2830	. . . . . 6 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → 𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))))
27		cnima 23150	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (𝑘Gen‘(𝐽 qTop 𝐹))) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (◡𝐹 “ 𝑥) ∈ 𝐽)
28	26, 27	sylancom 588	. . . . 5 ⊢ (((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝑘Gen‘(𝐽 qTop 𝐹))) → (◡𝐹 “ 𝑥) ∈ 𝐽)
29	2	elqtop2 23586	. . . . . 6 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
30	12, 16, 29	syl2anc 584	. . . . 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 713	. . . 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 3941	. 2 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹))
34		iskgen2 23433	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ ran 𝑘Gen ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ (𝑘Gen‘(𝐽 qTop 𝐹)) ⊆ (𝐽 qTop 𝐹)))
35	4, 33, 34	sylanbrc 583	1 ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ ran 𝑘Gen)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3903  ∪ cuni 4858  ^◡ccnv 5618  ran crn 5620   “ cima 5622   Fn wfn 6477  –onto→wfo 6480  ‘cfv 6482  (class class class)co 7349   qTop cqtop 17407  Topctop 22778  TopOnctopon 22795   Cn ccn 23109  𝑘Genckgen 23418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-1o 8388  df-map 8755  df-en 8873  df-dom 8874  df-fin 8876  df-fi 9301  df-rest 17326  df-topgen 17347  df-qtop 17411  df-top 22779  df-topon 22796  df-bases 22831  df-cn 23112  df-cmp 23272  df-kgen 23419

This theorem is referenced by: (None)