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Mirrors > Home > MPE Home > Th. List > setchom | Structured version Visualization version GIF version |
Description: Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
setcbas.c | ⊢ 𝐶 = (SetCat‘𝑈) |
setcbas.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
setchomfval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
setchom.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
setchom.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
Ref | Expression |
---|---|
setchom | ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑌 ↑𝑚 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setcbas.c | . . 3 ⊢ 𝐶 = (SetCat‘𝑈) | |
2 | setcbas.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | setchomfval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | 1, 2, 3 | setchomfval 17088 | . 2 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑𝑚 𝑥))) |
5 | simprr 789 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
6 | simprl 787 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋) | |
7 | 5, 6 | oveq12d 6928 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑦 ↑𝑚 𝑥) = (𝑌 ↑𝑚 𝑋)) |
8 | setchom.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
9 | setchom.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
10 | ovexd 6944 | . 2 ⊢ (𝜑 → (𝑌 ↑𝑚 𝑋) ∈ V) | |
11 | 4, 7, 8, 9, 10 | ovmpt2d 7053 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑌 ↑𝑚 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ‘cfv 6127 (class class class)co 6910 ↑𝑚 cmap 8127 Hom chom 16323 SetCatcsetc 17084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-hom 16336 df-cco 16337 df-setc 17085 |
This theorem is referenced by: elsetchom 17090 resssetc 17101 funcestrcsetclem8 17147 funcsetcestrclem8 17162 funcsetcestrclem9 17163 fthsetcestrc 17165 fullsetcestrc 17166 funcringcsetcALTV2lem8 42904 funcringcsetclem8ALTV 42927 |
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