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Theorem xkotf 21610
Description: Functionality of function 𝑇. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
xkoval.x 𝑋 = 𝑅
xkoval.k 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}
xkoval.t 𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
Assertion
Ref Expression
xkotf 𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
Distinct variable groups:   𝑣,𝑘,𝐾   𝑓,𝑘,𝑣,𝑥,𝑅   𝑆,𝑓,𝑘,𝑣,𝑥   𝑘,𝑋,𝑥
Allowed substitution hints:   𝑇(𝑥,𝑣,𝑓,𝑘)   𝐾(𝑥,𝑓)   𝑋(𝑣,𝑓)

Proof of Theorem xkotf
StepHypRef Expression
1 ssrab2 3828 . . . 4 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ⊆ (𝑅 Cn 𝑆)
2 ovex 6842 . . . . 5 (𝑅 Cn 𝑆) ∈ V
32elpw2 4977 . . . 4 ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆) ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ⊆ (𝑅 Cn 𝑆))
41, 3mpbir 221 . . 3 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆)
54rgen2w 3063 . 2 𝑘𝐾𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆)
6 xkoval.t . . 3 𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
76fmpt2 7406 . 2 (∀𝑘𝐾𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆) ↔ 𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆))
85, 7mpbi 220 1 𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  wral 3050  {crab 3054  wss 3715  𝒫 cpw 4302   cuni 4588   × cxp 5264  cima 5269  wf 6045  (class class class)co 6814  cmpt2 6816  t crest 16303   Cn ccn 21250  Compccmp 21411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335
This theorem is referenced by:  xkoopn  21614  xkouni  21624  xkoccn  21644  xkoco1cn  21682  xkoco2cn  21683  xkococn  21685  xkoinjcn  21712
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