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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 7434 | . 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 17921 | . . . . 5 ⊢ (𝜑 → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) |
| 7 | 6 | fveq2d 6910 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐼) = (2nd ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉)) |
| 8 | 3 | fvexi 6920 | . . . . . 6 ⊢ 𝐵 ∈ V |
| 9 | resiexg 7934 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
| 10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝐵) ∈ V |
| 11 | 8, 8 | xpex 7773 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
| 12 | 11 | mptex 7243 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V |
| 13 | 10, 12 | op2nd 8023 | . . . 4 ⊢ (2nd ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) |
| 14 | 7, 13 | eqtrdi 2793 | . . 3 ⊢ (𝜑 → (2nd ‘𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))) |
| 15 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → 𝑧 = 〈𝑋, 𝑌〉) | |
| 16 | 15 | fveq2d 6910 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐻‘𝑧) = (𝐻‘〈𝑋, 𝑌〉)) |
| 17 | df-ov 7434 | . . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
| 18 | 16, 17 | eqtr4di 2795 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐻‘𝑧) = (𝑋𝐻𝑌)) |
| 19 | 18 | reseq2d 5997 | . . 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 5724 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
| 23 | ovex 7464 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
| 24 | resiexg 7934 | . . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V) | |
| 25 | 23, 24 | mp1i 13 | . . 3 ⊢ (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V) |
| 26 | 14, 19, 22, 25 | fvmptd 7023 | . 2 ⊢ (𝜑 → ((2nd ‘𝐼)‘〈𝑋, 𝑌〉) = ( I ↾ (𝑋𝐻𝑌))) |
| 27 | 1, 26 | eqtrid 2789 | 1 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↦ cmpt 5225 I cid 5577 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 2nd c2nd 8013 Basecbs 17247 Hom chom 17308 Catccat 17707 idfunccidfu 17900 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-2nd 8015 df-idfu 17904 |
| This theorem is referenced by: idfu2 17923 idfucl 17926 cofulid 17935 cofurid 17936 idffth 17980 ressffth 17985 catciso 18156 |
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