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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 17822 | . . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌))) |
8 | 7 | fveq1d 6889 | . 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝐹) = (( I ↾ (𝑋𝐻𝑌))‘𝐹)) |
9 | idfu2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
10 | fvresi 7165 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → (( I ↾ (𝑋𝐻𝑌))‘𝐹) = 𝐹) |
12 | 8, 11 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 I cid 5571 ↾ cres 5676 ‘cfv 6539 (class class class)co 7403 2nd c2nd 7968 Basecbs 17139 Hom chom 17203 Catccat 17603 idfunccidfu 17800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-2nd 7970 df-idfu 17804 |
This theorem is referenced by: idfucl 17826 |
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