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Theorem xkotf 23594
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 ovex 7465 . . . 4 (𝑅 Cn 𝑆) ∈ V
2 ssrab2 4079 . . . 4 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ⊆ (𝑅 Cn 𝑆)
31, 2elpwi2 5334 . . 3 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆)
43rgen2w 3065 . 2 𝑘𝐾𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆)
5 xkoval.t . . 3 𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
65fmpo 8094 . 2 (∀𝑘𝐾𝑣𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆) ↔ 𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆))
74, 6mpbi 230 1 𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479  wss 3950  𝒫 cpw 4599   cuni 4906   × cxp 5682  cima 5687  wf 6556  (class class class)co 7432  cmpo 7434  t crest 17466   Cn ccn 23233  Compccmp 23395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016
This theorem is referenced by:  xkoopn  23598  xkouni  23608  xkoccn  23628  xkoco1cn  23666  xkoco2cn  23667  xkococn  23669  xkoinjcn  23696
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