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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 16662 | . . 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 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝐶) ∈ V | |
8 | 2, 7 | eqeltri 2684 | . . . . 5 ⊢ 𝐵 ∈ V |
9 | 8 | mptex 6390 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉) ∈ V |
10 | 8, 8 | mpt2ex 7136 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ 〈((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)〉)) ∈ V |
11 | 9, 10 | op1std 7069 | . . 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)〉)) |
12 | 6, 11 | syl 17 | . 2 ⊢ (𝜑 → (1st ‘𝑃) = (𝑥 ∈ 𝐵 ↦ 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉)) |
13 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
14 | 13 | fveq2d 6107 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋)) |
15 | 13 | fveq2d 6107 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑋)) |
16 | 14, 15 | opeq12d 4348 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
17 | prf1.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
18 | opex 4859 | . . 3 ⊢ 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 ∈ V | |
19 | 18 | a1i 11 | . 2 ⊢ (𝜑 → 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 ∈ V) |
20 | 12, 16, 17, 19 | fvmptd 6197 | 1 ⊢ (𝜑 → ((1st ‘𝑃)‘𝑋) = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 Basecbs 15695 Hom chom 15779 Func cfunc 16337 〈,〉F cprf 16634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-ixp 7795 df-func 16341 df-prf 16638 |
This theorem is referenced by: prfcl 16666 uncf1 16699 uncf2 16700 yonedalem21 16736 yonedalem22 16741 |
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