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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 7407 | . 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 17833 | . . . . 5 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩) |
7 | 6 | fveq2d 6888 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)) |
8 | 3 | fvexi 6898 | . . . . . 6 ⊢ 𝐵 ∈ V |
9 | resiexg 7901 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝐵) ∈ V |
11 | 8, 8 | xpex 7736 | . . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V |
12 | 11 | mptex 7219 | . . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V |
13 | 10, 12 | op2nd 7980 | . . . 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 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩) | |
16 | 15 | fveq2d 6888 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩)) |
17 | df-ov 7407 | . . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩) | |
18 | 16, 17 | eqtr4di 2784 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌)) |
19 | 18 | reseq2d 5974 | . . 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 5708 | . . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵)) |
23 | ovex 7437 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
24 | resiexg 7901 | . . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V) | |
25 | 23, 24 | mp1i 13 | . . 3 ⊢ (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V) |
26 | 14, 19, 22, 25 | fvmptd 6998 | . 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 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟨cop 4629 ↦ cmpt 5224 I cid 5566 × cxp 5667 ↾ cres 5671 ‘cfv 6536 (class class class)co 7404 2nd c2nd 7970 Basecbs 17151 Hom chom 17215 Catccat 17615 idfunccidfu 17812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-2nd 7972 df-idfu 17816 |
This theorem is referenced by: idfu2 17835 idfucl 17838 cofulid 17847 cofurid 17848 idffth 17893 ressffth 17898 catciso 18071 |
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