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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 7010 | . 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 16963 | . . . . 5 ⊢ (𝜑 → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) |
7 | 6 | fveq2d 6534 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐼) = (2nd ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉)) |
8 | 3 | fvexi 6544 | . . . . . 6 ⊢ 𝐵 ∈ V |
9 | resiexg 7466 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝐵) ∈ V |
11 | 8, 8 | xpex 7324 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
12 | 11 | mptex 6843 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V |
13 | 10, 12 | op2nd 7545 | . . . 4 ⊢ (2nd ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) |
14 | 7, 13 | syl6eq 2845 | . . 3 ⊢ (𝜑 → (2nd ‘𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))) |
15 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → 𝑧 = 〈𝑋, 𝑌〉) | |
16 | 15 | fveq2d 6534 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐻‘𝑧) = (𝐻‘〈𝑋, 𝑌〉)) |
17 | df-ov 7010 | . . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
18 | 16, 17 | syl6eqr 2847 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 〈𝑋, 𝑌〉) → (𝐻‘𝑧) = (𝑋𝐻𝑌)) |
19 | 18 | reseq2d 5726 | . . 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 5473 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
23 | ovex 7039 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
24 | resiexg 7466 | . . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V) | |
25 | 23, 24 | mp1i 13 | . . 3 ⊢ (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V) |
26 | 14, 19, 22, 25 | fvmptd 6632 | . 2 ⊢ (𝜑 → ((2nd ‘𝐼)‘〈𝑋, 𝑌〉) = ( I ↾ (𝑋𝐻𝑌))) |
27 | 1, 26 | syl5eq 2841 | 1 ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 Vcvv 3432 〈cop 4472 ↦ cmpt 5035 I cid 5339 × cxp 5433 ↾ cres 5437 ‘cfv 6217 (class class class)co 7007 2nd c2nd 7535 Basecbs 16300 Hom chom 16393 Catccat 16752 idfunccidfu 16942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-ov 7010 df-2nd 7537 df-idfu 16946 |
This theorem is referenced by: idfu2 16965 idfucl 16968 cofulid 16977 cofurid 16978 idffth 17020 ressffth 17025 catciso 17184 |
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