Theorem xkobval 22197

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 7287	. 2 ⊢ ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝐾 ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}}
3		oveq2 7167	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝑅 ↾t 𝑥) = (𝑅 ↾t 𝑘))
4	3	eleq1d 2900	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝑅 ↾t 𝑥) ∈ Comp ↔ (𝑅 ↾t 𝑘) ∈ Comp))
5	4	rexrab 3690	. . . 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 3417	. . . 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 3353	. . . . 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 3250	. . . 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 305	. . 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 2889	. 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 2847	1 ⊢ ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})}

Syntax hints:   ∧ wa 398   = wceq 1536   ∈ wcel 2113  {cab 2802  ∃wrex 3142  {crab 3145   ⊆ wss 3939  𝒫 cpw 4542  ∪ cuni 4841  ran crn 5559   “ cima 5561  (class class class)co 7159   ∈ cmpo 7161   ↾_t crest 16697   Cn ccn 21835  Compccmp 21997

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-cnv 5566  df-dm 5568  df-rn 5569  df-iota 6317  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164

This theorem is referenced by:  xkoccn  22230  xkoco1cn  22268  xkoco2cn  22269  xkoinjcn  22298