![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7419 | . 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 17890 | . . . . 5 ⊢ (𝜑 → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) |
7 | 6 | fveq2d 6897 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐼) = (2nd ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉)) |
8 | 3 | fvexi 6907 | . . . . . 6 ⊢ 𝐵 ∈ V |
9 | resiexg 7917 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝐵) ∈ V |
11 | 8, 8 | xpex 7753 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
12 | 11 | mptex 7232 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V |
13 | 10, 12 | op2nd 8004 | . . . 4 ⊢ (2nd ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) |
14 | 7, 13 | eqtrdi 2782 | . . 3 ⊢ (𝜑 → (2nd ‘𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))) |
15 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → 𝑧 = 〈𝑋, 𝑌〉) | |
16 | 15 | fveq2d 6897 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐻‘𝑧) = (𝐻‘〈𝑋, 𝑌〉)) |
17 | df-ov 7419 | . . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
18 | 16, 17 | eqtr4di 2784 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐻‘𝑧) = (𝑋𝐻𝑌)) |
19 | 18 | reseq2d 5981 | . . 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 5713 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
23 | ovex 7449 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
24 | resiexg 7917 | . . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V) | |
25 | 23, 24 | mp1i 13 | . . 3 ⊢ (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V) |
26 | 14, 19, 22, 25 | fvmptd 7008 | . 2 ⊢ (𝜑 → ((2nd ‘𝐼)‘〈𝑋, 𝑌〉) = ( I ↾ (𝑋𝐻𝑌))) |
27 | 1, 26 | eqtrid 2778 | 1 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 〈cop 4629 ↦ cmpt 5228 I cid 5571 × cxp 5672 ↾ cres 5676 ‘cfv 6546 (class class class)co 7416 2nd c2nd 7994 Basecbs 17208 Hom chom 17272 Catccat 17672 idfunccidfu 17869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 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 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-2nd 7996 df-idfu 17873 |
This theorem is referenced by: idfu2 17892 idfucl 17895 cofulid 17904 cofurid 17905 idffth 17950 ressffth 17955 catciso 18128 |
Copyright terms: Public domain | W3C validator |