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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 18092 | . . 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 6857 | . . . . 5 ⊢ 𝐵 ∈ V |
8 | 7 | mptex 7174 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) ∈ V |
9 | 7, 7 | mpoex 8013 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩)) ∈ V |
10 | 8, 9 | op1std 7932 | . . 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 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
13 | 12 | fveq2d 6847 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋)) |
14 | 12 | fveq2d 6847 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑋)) |
15 | 13, 14 | opeq12d 4839 | . 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 5422 | . . 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 6956 | 1 ⊢ (𝜑 → ((1st ‘𝑃)‘𝑋) = ⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⟨cop 4593 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 1st c1st 7920 2nd c2nd 7921 Basecbs 17088 Hom chom 17149 Func cfunc 17745 ⟨,⟩F cprf 18064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 df-ixp 8839 df-func 17749 df-prf 18068 |
This theorem is referenced by: prfcl 18096 uncf1 18130 uncf2 18131 yonedalem21 18167 yonedalem22 18172 |
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