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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 18093 | . 2 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑m 𝑥))) |
5 | simprr 771 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
6 | simprl 769 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋) | |
7 | 5, 6 | oveq12d 7431 | . 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 7448 | . 2 ⊢ (𝜑 → (𝑌 ↑m 𝑋) ∈ V) | |
11 | 4, 7, 8, 9, 10 | ovmpod 7567 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑌 ↑m 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ‘cfv 6543 (class class class)co 7413 ↑m cmap 8844 Hom chom 17269 SetCatcsetc 18089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12516 df-z 12602 df-dec 12721 df-uz 12866 df-fz 13530 df-struct 17141 df-slot 17176 df-ndx 17188 df-base 17206 df-hom 17282 df-cco 17283 df-setc 18090 |
This theorem is referenced by: elsetchom 18095 resssetc 18106 funcestrcsetclem8 18163 funcsetcestrclem8 18178 funcsetcestrclem9 18179 fthsetcestrc 18181 fullsetcestrc 18182 funcringcsetcALTV2lem8 47707 funcringcsetclem8ALTV 47730 |
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