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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 16856 | . . . 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 794 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶) | |
5 | simprr 796 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷) | |
6 | 4, 5 | opeq12d 4410 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 〈𝑐, 𝑑〉 = 〈𝐶, 𝐷〉) |
7 | 4, 5 | oveq12d 6668 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 1stF 𝑑) = (𝐶 1stF 𝐷)) |
8 | 6, 7 | oveq12d 6668 | . . 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 6680 | . . 3 ⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) ∈ V) | |
12 | 3, 8, 9, 10, 11 | ovmpt2d 6788 | . 2 ⊢ (𝜑 → (𝐶Δfunc𝐷) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
13 | 1, 12 | syl5eq 2668 | 1 ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 (class class class)co 6650 ↦ cmpt2 6652 Catccat 16325 1stF c1stf 16809 curryF ccurf 16850 Δfunccdiag 16852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-diag 16856 |
This theorem is referenced by: diagcl 16881 diag11 16883 diag12 16884 diag2 16885 |
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