Theorem prf2 17835

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 17834	. 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 6760	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐹)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾))
11	9	fveq2d 6760	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐺)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾))
12	10, 11	opeq12d 4809	. 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 5373	. . 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 6864	1 ⊢ (𝜑 → ((𝑋(2nd ‘𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255  2^nd c2nd 7803  Basecbs 16840  Hom chom 16899   Func cfunc 17485   ⟨,⟩_F cprf 17804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-ixp 8644  df-func 17489  df-prf 17808

This theorem is referenced by:  prfcl  17836  prf1st  17837  prf2nd  17838  uncf2  17871  yonedalem22  17912