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| Mirrors > Home > MPE Home > Th. List > diagval | Structured version Visualization version GIF version | ||
| Description: Define the diagonal functor, which is the functor 𝐶⟶(𝐷 Func 𝐶) whose object part is 𝑥 ∈ 𝐶 ↦ (𝑦 ∈ 𝐷 ↦ 𝑥). We can define this equationally as the currying of the first projection functor, and by expressing it this way we get a quick proof of functoriality. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.) |
| Ref | Expression |
|---|---|
| diagval.l | ⊢ 𝐿 = (𝐶Δfunc𝐷) |
| diagval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
| diagval.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
| Ref | Expression |
|---|---|
| diagval | ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | diagval.l | . 2 ⊢ 𝐿 = (𝐶Δfunc𝐷) | |
| 2 | df-diag 18261 | . . . 4 ⊢ Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (〈𝑐, 𝑑〉 curryF (𝑐 1stF 𝑑))) | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (〈𝑐, 𝑑〉 curryF (𝑐 1stF 𝑑)))) |
| 4 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶) | |
| 5 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷) | |
| 6 | 4, 5 | opeq12d 4881 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 〈𝑐, 𝑑〉 = 〈𝐶, 𝐷〉) |
| 7 | 4, 5 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 1stF 𝑑) = (𝐶 1stF 𝐷)) |
| 8 | 6, 7 | oveq12d 7449 | . . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (〈𝑐, 𝑑〉 curryF (𝑐 1stF 𝑑)) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
| 9 | diagval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 10 | diagval.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
| 11 | ovexd 7466 | . . 3 ⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) ∈ V) | |
| 12 | 3, 8, 9, 10, 11 | ovmpod 7585 | . 2 ⊢ (𝜑 → (𝐶Δfunc𝐷) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
| 13 | 1, 12 | eqtrid 2789 | 1 ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 (class class class)co 7431 ∈ cmpo 7433 Catccat 17707 1stF c1stf 18214 curryF ccurf 18255 Δfunccdiag 18257 |
| 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-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-diag 18261 |
| This theorem is referenced by: diagcl 18286 diag11 18288 diag12 18289 diag2 18290 |
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