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Theorem xkotf 21436
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 3720 . . . 4 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ⊆ (𝑅 Cn 𝑆)
2 ovex 6718 . . . . 5 (𝑅 Cn 𝑆) ∈ V
32elpw2 4858 . . . 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 2954 . 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 7282 . 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 1523  wcel 2030  wral 2941  {crab 2945  wss 3607  𝒫 cpw 4191   cuni 4468   × cxp 5141  cima 5146  wf 5922  (class class class)co 6690  cmpt2 6692  t crest 16128   Cn ccn 21076  Compccmp 21237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211
This theorem is referenced by:  xkoopn  21440  xkouni  21450  xkoccn  21470  xkoco1cn  21508  xkoco2cn  21509  xkococn  21511  xkoinjcn  21538
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