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Mirrors > Home > MPE Home > Th. List > prf2 | Structured version Visualization version GIF version |
Description: Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
prfval.k | ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) |
prfval.b | ⊢ 𝐵 = (Base‘𝐶) |
prfval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
prfval.c | ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
prfval.d | ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) |
prf1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
prf2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
prf2.k | ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) |
Ref | Expression |
---|---|
prf2 | ⊢ (𝜑 → ((𝑋(2nd ‘𝑃)𝑌)‘𝐾) = 〈((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prfval.k | . . 3 ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) | |
2 | prfval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
3 | prfval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | prfval.c | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) | |
5 | prfval.d | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) | |
6 | prf1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | prf2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | prf2fval 17929 | . 2 ⊢ (𝜑 → (𝑋(2nd ‘𝑃)𝑌) = (ℎ ∈ (𝑋𝐻𝑌) ↦ 〈((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)〉)) |
9 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ℎ = 𝐾) | |
10 | 9 | fveq2d 6775 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐹)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾)) |
11 | 9 | fveq2d 6775 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐺)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)) |
12 | 10, 11 | opeq12d 4818 | . 2 ⊢ ((𝜑 ∧ ℎ = 𝐾) → 〈((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)〉 = 〈((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)〉) |
13 | prf2.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) | |
14 | opex 5383 | . . 3 ⊢ 〈((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)〉 ∈ V | |
15 | 14 | a1i 11 | . 2 ⊢ (𝜑 → 〈((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)〉 ∈ V) |
16 | 8, 12, 13, 15 | fvmptd 6879 | 1 ⊢ (𝜑 → ((𝑋(2nd ‘𝑃)𝑌)‘𝐾) = 〈((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 〈cop 4573 ‘cfv 6432 (class class class)co 7272 2nd c2nd 7824 Basecbs 16923 Hom chom 16984 Func cfunc 17580 〈,〉F cprf 17899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7275 df-oprab 7276 df-mpo 7277 df-1st 7825 df-2nd 7826 df-map 8609 df-ixp 8678 df-func 17584 df-prf 17903 |
This theorem is referenced by: prfcl 17931 prf1st 17932 prf2nd 17933 uncf2 17966 yonedalem22 18007 |
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