prf1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > prf1		Structured version Visualization version GIF version

Theorem prf1 18374

Description: Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
prf1	⊢ (𝜑 → ((1^st ‘𝑃)‘𝑋) = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)

Proof of Theorem prf1 Dummy variables ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

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 18373	. . 3 ⊢ (𝜑 → 𝑃 = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2^nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)
7	2	fvexi 6899	. . . . 5 ⊢ 𝐵 ∈ V
8	7	mptex 7229	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩) ∈ V
9	7, 7	mpoex 8092	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2^nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)⟩)) ∈ V
10	8, 9	op1std 8011	. . 3 ⊢ (𝑃 = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2^nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)⟩))⟩ → (1^st ‘𝑃) = (𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩))
11	6, 10	syl 18	. 2 ⊢ (𝜑 → (1^st ‘𝑃) = (𝑥 ∈ 𝐵 ↦ ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩))
12		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
13	12	fveq2d 6889	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑋))
14	12	fveq2d 6889	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑋))
15	13, 14	opeq12d 4840	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)
16		prf1.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17		opex 5431	. . 3 ⊢ ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩ ∈ V
18	17	a1i 11	. 2 ⊢ (𝜑 → ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩ ∈ V)
19	11, 15, 16, 18	fvmptd 7001	1 ⊢ (𝜑 → ((1^st ‘𝑃)‘𝑋) = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⟨cop 4589 ↦ cmpt 5185 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7999 2^nd c2nd 8000 Basecbs 17387 Hom chom 17439 Func cfunc 18029 ⟨,⟩_F cprf 18345

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-map 8849 df-ixp 8926 df-func 18033 df-prf 18349

This theorem is used by: prfcl 18377 uncf1 18410 uncf2 18411 yonedalem21 18447 yonedalem22 18452

W3C validator