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Mirrors > Home > MPE Home > Th. List > setcco | Structured version Visualization version GIF version |
Description: Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
setcbas.c | ⊢ 𝐶 = (SetCat‘𝑈) |
setcbas.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
setcco.o | ⊢ · = (comp‘𝐶) |
setcco.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
setcco.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
setcco.z | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
setcco.f | ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
setcco.g | ⊢ (𝜑 → 𝐺:𝑌⟶𝑍) |
Ref | Expression |
---|---|
setcco | ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setcbas.c | . . . 4 ⊢ 𝐶 = (SetCat‘𝑈) | |
2 | setcbas.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | setcco.o | . . . 4 ⊢ · = (comp‘𝐶) | |
4 | 1, 2, 3 | setccofval 18136 | . . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) |
5 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
6 | simprl 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑣 = 〈𝑋, 𝑌〉) | |
7 | 6 | fveq2d 6911 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = (2nd ‘〈𝑋, 𝑌〉)) |
8 | setcco.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
9 | setcco.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
10 | op2ndg 8026 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
11 | 8, 9, 10 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
13 | 7, 12 | eqtrd 2775 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌) |
14 | 5, 13 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑧 ↑m (2nd ‘𝑣)) = (𝑍 ↑m 𝑌)) |
15 | 6 | fveq2d 6911 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘〈𝑋, 𝑌〉)) |
16 | op1stg 8025 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
17 | 8, 9, 16 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
18 | 17 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
19 | 15, 18 | eqtrd 2775 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋) |
20 | 13, 19 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = (𝑌 ↑m 𝑋)) |
21 | eqidd 2736 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓)) | |
22 | 14, 20, 21 | mpoeq123dv 7508 | . . 3 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓))) |
23 | 8, 9 | opelxpd 5728 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝑈 × 𝑈)) |
24 | setcco.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
25 | ovex 7464 | . . . . 5 ⊢ (𝑍 ↑m 𝑌) ∈ V | |
26 | ovex 7464 | . . . . 5 ⊢ (𝑌 ↑m 𝑋) ∈ V | |
27 | 25, 26 | mpoex 8103 | . . . 4 ⊢ (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V) |
29 | 4, 22, 23, 24, 28 | ovmpod 7585 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓))) |
30 | simprl 771 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺) | |
31 | simprr 773 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹) | |
32 | 30, 31 | coeq12d 5878 | . 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹)) |
33 | setcco.g | . . 3 ⊢ (𝜑 → 𝐺:𝑌⟶𝑍) | |
34 | 24, 9 | elmapd 8879 | . . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑍 ↑m 𝑌) ↔ 𝐺:𝑌⟶𝑍)) |
35 | 33, 34 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑍 ↑m 𝑌)) |
36 | setcco.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) | |
37 | 9, 8 | elmapd 8879 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌)) |
38 | 36, 37 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 ↑m 𝑋)) |
39 | coexg 7952 | . . 3 ⊢ ((𝐺 ∈ (𝑍 ↑m 𝑌) ∧ 𝐹 ∈ (𝑌 ↑m 𝑋)) → (𝐺 ∘ 𝐹) ∈ V) | |
40 | 35, 38, 39 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
41 | 29, 32, 35, 38, 40 | ovmpod 7585 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 × cxp 5687 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8011 2nd c2nd 8012 ↑m cmap 8865 compcco 17310 SetCatcsetc 18129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-hom 17322 df-cco 17323 df-setc 18130 |
This theorem is referenced by: setccatid 18138 setcmon 18141 setcepi 18142 setcsect 18143 resssetc 18146 funcestrcsetclem9 18204 funcsetcestrclem9 18219 hofcllem 18315 yonedalem4c 18334 yonedalem3b 18336 yonedainv 18338 funcringcsetcALTV2lem9 48142 funcringcsetclem9ALTV 48165 |
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