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Mirrors > Home > MPE Home > Th. List > idfu2nd | Structured version Visualization version GIF version |
Description: Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-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) |
Ref | Expression |
---|---|
idfu2nd | ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7159 | . 2 ⊢ (𝑋(2nd ‘𝐼)𝑌) = ((2nd ‘𝐼)‘〈𝑋, 𝑌〉) | |
2 | idfuval.i | . . . . . 6 ⊢ 𝐼 = (idfunc‘𝐶) | |
3 | idfuval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
4 | idfuval.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | idfuval.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | 2, 3, 4, 5 | idfuval 17146 | . . . . 5 ⊢ (𝜑 → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) |
7 | 6 | fveq2d 6674 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐼) = (2nd ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉)) |
8 | 3 | fvexi 6684 | . . . . . 6 ⊢ 𝐵 ∈ V |
9 | resiexg 7619 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝐵) ∈ V |
11 | 8, 8 | xpex 7476 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
12 | 11 | mptex 6986 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V |
13 | 10, 12 | op2nd 7698 | . . . 4 ⊢ (2nd ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) |
14 | 7, 13 | syl6eq 2872 | . . 3 ⊢ (𝜑 → (2nd ‘𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))) |
15 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → 𝑧 = 〈𝑋, 𝑌〉) | |
16 | 15 | fveq2d 6674 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐻‘𝑧) = (𝐻‘〈𝑋, 𝑌〉)) |
17 | df-ov 7159 | . . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
18 | 16, 17 | syl6eqr 2874 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐻‘𝑧) = (𝑋𝐻𝑌)) |
19 | 18 | reseq2d 5853 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → ( I ↾ (𝐻‘𝑧)) = ( I ↾ (𝑋𝐻𝑌))) |
20 | idfu2nd.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
21 | idfu2nd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
22 | 20, 21 | opelxpd 5593 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
23 | ovex 7189 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
24 | resiexg 7619 | . . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V) | |
25 | 23, 24 | mp1i 13 | . . 3 ⊢ (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V) |
26 | 14, 19, 22, 25 | fvmptd 6775 | . 2 ⊢ (𝜑 → ((2nd ‘𝐼)‘〈𝑋, 𝑌〉) = ( I ↾ (𝑋𝐻𝑌))) |
27 | 1, 26 | syl5eq 2868 | 1 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 ↦ cmpt 5146 I cid 5459 × cxp 5553 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 2nd c2nd 7688 Basecbs 16483 Hom chom 16576 Catccat 16935 idfunccidfu 17125 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 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-ral 3143 df-rex 3144 df-reu 3145 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-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 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-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-2nd 7690 df-idfu 17129 |
This theorem is referenced by: idfu2 17148 idfucl 17151 cofulid 17160 cofurid 17161 idffth 17203 ressffth 17208 catciso 17367 |
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