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Mirrors > Home > MPE Home > Th. List > idfu2 | Structured version Visualization version GIF version |
Description: Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.) |
Ref | Expression |
---|---|
idfuval.i | ⊢ 𝐼 = (idfunc‘𝐶) |
idfuval.b | ⊢ 𝐵 = (Base‘𝐶) |
idfuval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
idfuval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
idfu2nd.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
idfu2nd.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
idfu2.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
Ref | Expression |
---|---|
idfu2 | ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝐹) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idfuval.i | . . . 4 ⊢ 𝐼 = (idfunc‘𝐶) | |
2 | idfuval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | idfuval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | idfuval.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
5 | idfu2nd.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | idfu2nd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | idfu2nd 16537 | . . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌))) |
8 | 7 | fveq1d 6193 | . 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝐹) = (( I ↾ (𝑋𝐻𝑌))‘𝐹)) |
9 | idfu2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
10 | fvresi 6439 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹) |
12 | 8, 11 | eqtrd 2656 | 1 ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 I cid 5023 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 2nd c2nd 7167 Basecbs 15857 Hom chom 15952 Catccat 16325 idfunccidfu 16515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-2nd 7169 df-idfu 16519 |
This theorem is referenced by: idfucl 16541 |
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