Theorem prf2 18157

Description: Value of the pairing functor on morphisms. (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)
prf2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
prf2.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
prf2	⊢ (𝜑 → ((𝑋(2nd ‘𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)⟩)

Proof of Theorem prf2

Dummy variable ℎ is distinct from all other variables.

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 18156	. 2 ⊢ (𝜑 → (𝑋(2nd ‘𝑃)𝑌) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)⟩))
9		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ℎ = 𝐾)
10	9	fveq2d 6833	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐹)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾))
11	9	fveq2d 6833	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐺)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾))
12	10, 11	opeq12d 4814	. 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 5405	. . 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 6944	1 ⊢ (𝜑 → ((𝑋(2nd ‘𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3427  ⟨cop 4563  ‘cfv 6487  (class class class)co 7356  2^nd c2nd 7930  Basecbs 17168  Hom chom 17220   Func cfunc 17810   ⟨,⟩_F cprf 18126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8764  df-ixp 8835  df-func 17814  df-prf 18130

This theorem is referenced by:  prfcl  18158  prf1st  18159  prf2nd  18160  uncf2  18192  yonedalem22  18233