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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 17453 | . 2 ⊢ (𝜑 → (𝑋(2nd ‘𝑃)𝑌) = (ℎ ∈ (𝑋𝐻𝑌) ↦ 〈((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)〉)) |
9 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ℎ = 𝐾) | |
10 | 9 | fveq2d 6676 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐹)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾)) |
11 | 9 | fveq2d 6676 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐺)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)) |
12 | 10, 11 | opeq12d 4813 | . 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 5358 | . . 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 6777 | 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 ‘cfv 6357 (class class class)co 7158 2nd c2nd 7690 Basecbs 16485 Hom chom 16578 Func cfunc 17126 〈,〉F cprf 17423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-ixp 8464 df-func 17130 df-prf 17427 |
This theorem is referenced by: prfcl 17455 prf1st 17456 prf2nd 17457 uncf2 17489 yonedalem22 17530 |
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