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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 16555 | . . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑𝑚 (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑𝑚 (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) |
5 | simprr 792 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
6 | simprl 790 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑣 = 〈𝑋, 𝑌〉) | |
7 | 6 | fveq2d 6107 | . . . . . 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 7072 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
11 | 8, 9, 10 | syl2anc 691 | . . . . . . 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 2644 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌) |
14 | 5, 13 | oveq12d 6567 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑧 ↑𝑚 (2nd ‘𝑣)) = (𝑍 ↑𝑚 𝑌)) |
15 | 6 | fveq2d 6107 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘〈𝑋, 𝑌〉)) |
16 | op1stg 7071 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
17 | 8, 9, 16 | syl2anc 691 | . . . . . . 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 2644 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋) |
20 | 13, 19 | oveq12d 6567 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) ↑𝑚 (1st ‘𝑣)) = (𝑌 ↑𝑚 𝑋)) |
21 | eqidd 2611 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓)) | |
22 | 14, 20, 21 | mpt2eq123dv 6615 | . . 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 | opelxpi 5072 | . . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → 〈𝑋, 𝑌〉 ∈ (𝑈 × 𝑈)) | |
24 | 8, 9, 23 | syl2anc 691 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝑈 × 𝑈)) |
25 | setcco.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
26 | ovex 6577 | . . . . 5 ⊢ (𝑍 ↑𝑚 𝑌) ∈ V | |
27 | ovex 6577 | . . . . 5 ⊢ (𝑌 ↑𝑚 𝑋) ∈ V | |
28 | 26, 27 | mpt2ex 7136 | . . . 4 ⊢ (𝑔 ∈ (𝑍 ↑𝑚 𝑌), 𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V |
29 | 28 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑍 ↑𝑚 𝑌), 𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V) |
30 | 4, 22, 24, 25, 29 | ovmpt2d 6686 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (𝑔 ∈ (𝑍 ↑𝑚 𝑌), 𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↦ (𝑔 ∘ 𝑓))) |
31 | simprl 790 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺) | |
32 | simprr 792 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹) | |
33 | 31, 32 | coeq12d 5208 | . 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹)) |
34 | setcco.g | . . 3 ⊢ (𝜑 → 𝐺:𝑌⟶𝑍) | |
35 | 25, 9 | elmapd 7758 | . . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑍 ↑𝑚 𝑌) ↔ 𝐺:𝑌⟶𝑍)) |
36 | 34, 35 | mpbird 246 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑍 ↑𝑚 𝑌)) |
37 | setcco.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) | |
38 | 9, 8 | elmapd 7758 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝐹:𝑋⟶𝑌)) |
39 | 37, 38 | mpbird 246 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 ↑𝑚 𝑋)) |
40 | coexg 7010 | . . 3 ⊢ ((𝐺 ∈ (𝑍 ↑𝑚 𝑌) ∧ 𝐹 ∈ (𝑌 ↑𝑚 𝑋)) → (𝐺 ∘ 𝐹) ∈ V) | |
41 | 36, 39, 40 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
42 | 30, 33, 36, 39, 41 | ovmpt2d 6686 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 × cxp 5036 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 ↑𝑚 cmap 7744 compcco 15780 SetCatcsetc 16548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-hom 15793 df-cco 15794 df-setc 16549 |
This theorem is referenced by: setccatid 16557 setcmon 16560 setcepi 16561 setcsect 16562 resssetc 16565 funcestrcsetclem9 16611 funcsetcestrclem9 16626 hofcllem 16721 yonedalem4c 16740 yonedalem3b 16742 yonedainv 16744 funcringcsetcALTV2lem9 41836 funcringcsetclem9ALTV 41859 |
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