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Mirrors > Home > MPE Home > Th. List > prf1 | Structured version Visualization version GIF version |
Description: Value of the pairing functor on objects. (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) |
Ref | Expression |
---|---|
prf1 | ⊢ (𝜑 → ((1st ‘𝑃)‘𝑋) = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prfval.k | . . . 4 ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) | |
2 | prfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | prfval.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | prfval.c | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) | |
5 | prfval.d | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) | |
6 | 1, 2, 3, 4, 5 | prfval 18197 | . . 3 ⊢ (𝜑 → 𝑃 = 〈(𝑥 ∈ 𝐵 ↦ 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ 〈((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)〉))〉) |
7 | 2 | fvexi 6916 | . . . . 5 ⊢ 𝐵 ∈ V |
8 | 7 | mptex 7241 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉) ∈ V |
9 | 7, 7 | mpoex 8090 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ 〈((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)〉)) ∈ V |
10 | 8, 9 | op1std 8009 | . . 3 ⊢ (𝑃 = 〈(𝑥 ∈ 𝐵 ↦ 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ 〈((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)〉))〉 → (1st ‘𝑃) = (𝑥 ∈ 𝐵 ↦ 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉)) |
11 | 6, 10 | syl 17 | . 2 ⊢ (𝜑 → (1st ‘𝑃) = (𝑥 ∈ 𝐵 ↦ 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉)) |
12 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
13 | 12 | fveq2d 6906 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋)) |
14 | 12 | fveq2d 6906 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑋)) |
15 | 13, 14 | opeq12d 4886 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
16 | prf1.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
17 | opex 5470 | . . 3 ⊢ 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 ∈ V | |
18 | 17 | a1i 11 | . 2 ⊢ (𝜑 → 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 ∈ V) |
19 | 11, 15, 16, 18 | fvmptd 7017 | 1 ⊢ (𝜑 → ((1st ‘𝑃)‘𝑋) = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 〈cop 4638 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 1st c1st 7997 2nd c2nd 7998 Basecbs 17187 Hom chom 17251 Func cfunc 17847 〈,〉F cprf 18169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 df-ixp 8923 df-func 17851 df-prf 18173 |
This theorem is referenced by: prfcl 18201 uncf1 18235 uncf2 18236 yonedalem21 18272 yonedalem22 18277 |
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