Theorem prf2 18214

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 18213	. 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 6880	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐹)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾))
11	9	fveq2d 6880	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐺)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾))
12	10, 11	opeq12d 4857	. 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 5439	. . 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 6993	1 ⊢ (𝜑 → ((𝑋(2nd ‘𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607  ‘cfv 6531  (class class class)co 7405  2^nd c2nd 7987  Basecbs 17228  Hom chom 17282   Func cfunc 17867   ⟨,⟩_F cprf 18183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-map 8842  df-ixp 8912  df-func 17871  df-prf 18187

This theorem is referenced by:  prfcl  18215  prf1st  18216  prf2nd  18217  uncf2  18249  yonedalem22  18290