Theorem xkobval 23521

Description: Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypotheses

Ref	Expression
xkoval.x	⊢ 𝑋 = ∪ 𝑅
xkoval.k	⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}
xkoval.t	⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})

Assertion

Ref	Expression
xkobval	⊢ ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})}

Distinct variable groups:   𝑘,𝑠,𝑣,𝐾   𝑓,𝑘,𝑠,𝑣,𝑥,𝑅   𝑆,𝑓,𝑘,𝑠,𝑣,𝑥   𝑇,𝑠   𝑘,𝑋,𝑥

Allowed substitution hints:   𝑇(𝑥,𝑣,𝑓,𝑘)   𝐾(𝑥,𝑓)   𝑋(𝑣,𝑓,𝑠)

Proof of Theorem xkobval

Step	Hyp	Ref	Expression
1		xkoval.t	. . 3 ⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
2	1	rnmpo 7488	. 2 ⊢ ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝐾 ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}}
3		oveq2 7363	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝑅 ↾t 𝑥) = (𝑅 ↾t 𝑘))
4	3	eleq1d 2818	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝑅 ↾t 𝑥) ∈ Comp ↔ (𝑅 ↾t 𝑘) ∈ Comp))
5	4	rexrab 3651	. . . 4 ⊢ (∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 𝑋((𝑅 ↾t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
6		xkoval.k	. . . . 5 ⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}
7	6	rexeqi 3292	. . . 4 ⊢ (∃𝑘 ∈ 𝐾 ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
8		r19.42v 3165	. . . . 5 ⊢ (∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ↔ ((𝑅 ↾t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
9	8	rexbii 3080	. . . 4 ⊢ (∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 𝑋((𝑅 ↾t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
10	5, 7, 9	3bitr4i 303	. . 3 ⊢ (∃𝑘 ∈ 𝐾 ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
11	10	abbii 2800	. 2 ⊢ {𝑠 ∣ ∃𝑘 ∈ 𝐾 ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})}
12	2, 11	eqtri 2756	1 ⊢ ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})}

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  ∃wrex 3057  {crab 3396   ⊆ wss 3898  𝒫 cpw 4551  ∪ cuni 4860  ran crn 5622   “ cima 5624  (class class class)co 7355   ∈ cmpo 7357   ↾_t crest 17331   Cn ccn 23159  Compccmp 23321

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-sep 5238  ax-nul 5248  ax-pr 5374

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-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-cnv 5629  df-dm 5631  df-rn 5632  df-iota 6445  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360

This theorem is referenced by:  xkoccn  23554  xkoco1cn  23592  xkoco2cn  23593  xkoinjcn  23622