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| Description: Functionality of function 𝑇. (Contributed by Mario Carneiro, 19-Mar-2015.) | 
| 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}) | 
| Ref | Expression | 
|---|---|
| xkotf | ⊢ 𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ovex 7465 | . . . 4 ⊢ (𝑅 Cn 𝑆) ∈ V | |
| 2 | ssrab2 4079 | . . . 4 ⊢ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ⊆ (𝑅 Cn 𝑆) | |
| 3 | 1, 2 | elpwi2 5334 | . . 3 ⊢ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆) | 
| 4 | 3 | rgen2w 3065 | . 2 ⊢ ∀𝑘 ∈ 𝐾 ∀𝑣 ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆) | 
| 5 | xkoval.t | . . 3 ⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) | |
| 6 | 5 | fmpo 8094 | . 2 ⊢ (∀𝑘 ∈ 𝐾 ∀𝑣 ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ∈ 𝒫 (𝑅 Cn 𝑆) ↔ 𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)) | 
| 7 | 4, 6 | mpbi 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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