Theorem prf2 18264

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 18263	. 2 ⊢ (𝜑 → (𝑋(2nd ‘𝑃)𝑌) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)⟩))
9		simpr 489	. . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ℎ = 𝐾)
10	9	fveq2d 6885	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐹)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾))
11	9	fveq2d 6885	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝑋(2nd ‘𝐺)𝑌)‘ℎ) = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾))
12	10, 11	opeq12d 4845	. 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 5444	. . 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 6997	1 ⊢ (𝜑 → ((𝑋(2nd ‘𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454  ⟨cop 4594  ‘cfv 6536  (class class class)co 7412  2^nd c2nd 7983  Basecbs 17275  Hom chom 17327   Func cfunc 17917   ⟨,⟩_F cprf 18233

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-map 8824  df-ixp 8894  df-func 17921  df-prf 18237

This theorem is used by:  prfcl  18265  prf1st  18266  prf2nd  18267  uncf2  18299  yonedalem22  18340