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Mirrors > Home > MPE Home > Th. List > fucid | Structured version Visualization version GIF version |
Description: The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
fucid.q | ⊢ 𝑄 = (𝐶 FuncCat 𝐷) |
fucid.i | ⊢ 𝐼 = (Id‘𝑄) |
fucid.1 | ⊢ 1 = (Id‘𝐷) |
fucid.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
Ref | Expression |
---|---|
fucid | ⊢ (𝜑 → (𝐼‘𝐹) = ( 1 ∘ (1st ‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fucid.i | . . 3 ⊢ 𝐼 = (Id‘𝑄) | |
2 | fucid.q | . . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷) | |
3 | fucid.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) | |
4 | funcrcl 17133 | . . . . . . 7 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
6 | 5 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | 5 | simprd 498 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat) |
8 | fucid.1 | . . . . 5 ⊢ 1 = (Id‘𝐷) | |
9 | 2, 6, 7, 8 | fuccatid 17239 | . . . 4 ⊢ (𝜑 → (𝑄 ∈ Cat ∧ (Id‘𝑄) = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1st ‘𝑓))))) |
10 | 9 | simprd 498 | . . 3 ⊢ (𝜑 → (Id‘𝑄) = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1st ‘𝑓)))) |
11 | 1, 10 | syl5eq 2868 | . 2 ⊢ (𝜑 → 𝐼 = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1st ‘𝑓)))) |
12 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹) | |
13 | 12 | fveq2d 6674 | . . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (1st ‘𝑓) = (1st ‘𝐹)) |
14 | 13 | coeq2d 5733 | . 2 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ( 1 ∘ (1st ‘𝑓)) = ( 1 ∘ (1st ‘𝐹))) |
15 | 8 | fvexi 6684 | . . . 4 ⊢ 1 ∈ V |
16 | fvex 6683 | . . . 4 ⊢ (1st ‘𝐹) ∈ V | |
17 | 15, 16 | coex 7635 | . . 3 ⊢ ( 1 ∘ (1st ‘𝐹)) ∈ V |
18 | 17 | a1i 11 | . 2 ⊢ (𝜑 → ( 1 ∘ (1st ‘𝐹)) ∈ V) |
19 | 11, 14, 3, 18 | fvmptd 6775 | 1 ⊢ (𝜑 → (𝐼‘𝐹) = ( 1 ∘ (1st ‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ↦ cmpt 5146 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 Catccat 16935 Idccid 16936 Func cfunc 17124 FuncCat cfuc 17212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-hom 16589 df-cco 16590 df-cat 16939 df-cid 16940 df-func 17128 df-nat 17213 df-fuc 17214 |
This theorem is referenced by: fucsect 17242 evlfcl 17472 curfcl 17482 curfuncf 17488 curf2ndf 17497 |
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