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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 18218 | . 2 ⊢ (𝜑 → (𝑋(2nd ‘𝑃)𝑌) = (ℎ ∈ (𝑋𝐻𝑌) ↦ 〈((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)〉)) |
9 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ℎ = 𝐾) | |
10 | 9 | fveq2d 6895 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐹)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾)) |
11 | 9 | fveq2d 6895 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐺)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)) |
12 | 10, 11 | opeq12d 4880 | . 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 5461 | . . 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 7006 | 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 394 = wceq 1534 ∈ wcel 2099 Vcvv 3463 〈cop 4630 ‘cfv 6544 (class class class)co 7414 2nd c2nd 7992 Basecbs 17206 Hom chom 17270 Func cfunc 17866 〈,〉F cprf 18188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7993 df-2nd 7994 df-map 8847 df-ixp 8917 df-func 17870 df-prf 18192 |
This theorem is referenced by: prfcl 18220 prf1st 18221 prf2nd 18222 uncf2 18255 yonedalem22 18296 |
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