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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 17542 | . . 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 6699 | . . . . . 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 7752 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
11 | 8, 9, 10 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
12 | 11 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
13 | 7, 12 | eqtrd 2771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌) |
14 | 5, 13 | oveq12d 7209 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑧 ↑m (2nd ‘𝑣)) = (𝑍 ↑m 𝑌)) |
15 | 6 | fveq2d 6699 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘〈𝑋, 𝑌〉)) |
16 | op1stg 7751 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
17 | 8, 9, 16 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
18 | 17 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
19 | 15, 18 | eqtrd 2771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋) |
20 | 13, 19 | oveq12d 7209 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = (𝑌 ↑m 𝑋)) |
21 | eqidd 2737 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓)) | |
22 | 14, 20, 21 | mpoeq123dv 7264 | . . 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 5574 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝑈 × 𝑈)) |
24 | setcco.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
25 | ovex 7224 | . . . . 5 ⊢ (𝑍 ↑m 𝑌) ∈ V | |
26 | ovex 7224 | . . . . 5 ⊢ (𝑌 ↑m 𝑋) ∈ V | |
27 | 25, 26 | mpoex 7828 | . . . 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 7339 | . 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 5718 | . 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹)) |
33 | setcco.g | . . 3 ⊢ (𝜑 → 𝐺:𝑌⟶𝑍) | |
34 | 24, 9 | elmapd 8500 | . . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑍 ↑m 𝑌) ↔ 𝐺:𝑌⟶𝑍)) |
35 | 33, 34 | mpbird 260 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑍 ↑m 𝑌)) |
36 | setcco.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) | |
37 | 9, 8 | elmapd 8500 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌)) |
38 | 36, 37 | mpbird 260 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 ↑m 𝑋)) |
39 | coexg 7685 | . . 3 ⊢ ((𝐺 ∈ (𝑍 ↑m 𝑌) ∧ 𝐹 ∈ (𝑌 ↑m 𝑋)) → (𝐺 ∘ 𝐹) ∈ V) | |
40 | 35, 38, 39 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
41 | 29, 32, 35, 38, 40 | ovmpod 7339 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 〈cop 4533 × cxp 5534 ∘ ccom 5540 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 1st c1st 7737 2nd c2nd 7738 ↑m cmap 8486 compcco 16761 SetCatcsetc 17535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-hom 16773 df-cco 16774 df-setc 17536 |
This theorem is referenced by: setccatid 17544 setcmon 17547 setcepi 17548 setcsect 17549 resssetc 17552 funcestrcsetclem9 17609 funcsetcestrclem9 17624 hofcllem 17720 yonedalem4c 17739 yonedalem3b 17741 yonedainv 17743 funcringcsetcALTV2lem9 45218 funcringcsetclem9ALTV 45241 |
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