Theorem prf2fval 18295

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)

Assertion

Ref	Expression
prf2fval	⊢ (𝜑 → (𝑋(2nd ‘𝑃)𝑌) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)⟩))

Distinct variable groups:   𝐵,ℎ   ℎ,𝐹   𝜑,ℎ   ℎ,𝐺   ℎ,𝑋   ℎ,𝑌   ℎ,𝐻

Allowed substitution hints:   𝐶(ℎ)   𝐷(ℎ)   𝑃(ℎ)   𝐸(ℎ)

Proof of Theorem prf2fval

Dummy variables 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 18293	. . 3 ⊢ (𝜑 → 𝑃 = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)
7	2	fvexi 6896	. . . . 5 ⊢ 𝐵 ∈ V
8	7	mptex 7226	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) ∈ V
9	7, 7	mpoex 8082	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩)) ∈ V
10	8, 9	op2ndd 8001	. . 3 ⊢ (𝑃 = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩ → (2nd ‘𝑃) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩)))
11	6, 10	syl 18	. 2 ⊢ (𝜑 → (2nd ‘𝑃) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩)))
12		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
13		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
14	12, 13	oveq12d 7435	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
15	12, 13	oveq12d 7435	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥(2nd ‘𝐹)𝑦) = (𝑋(2nd ‘𝐹)𝑌))
16	15	fveq1d 6884	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑥(2nd ‘𝐹)𝑦)‘ℎ) = ((𝑋(2nd ‘𝐹)𝑌)‘ℎ))
17	12, 13	oveq12d 7435	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥(2nd ‘𝐺)𝑦) = (𝑋(2nd ‘𝐺)𝑌))
18	17	fveq1d 6884	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑥(2nd ‘𝐺)𝑦)‘ℎ) = ((𝑋(2nd ‘𝐺)𝑌)‘ℎ))
19	16, 18	opeq12d 4844	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩ = ⟨((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)⟩)
20	14, 19	mpteq12dv 5196	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)⟩))
21		prf1.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		prf2.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
23		ovex 7450	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
24	23	mptex 7226	. . 3 ⊢ (ℎ ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)⟩) ∈ V
25	24	a1i 11	. 2 ⊢ (𝜑 → (ℎ ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)⟩) ∈ V)
26	11, 20, 21, 22, 25	ovmpod 7569	1 ⊢ (𝜑 → (𝑋(2nd ‘𝑃)𝑌) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ⟨cop 4593   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  1^st c1st 7988  2^nd c2nd 7989  Basecbs 17307  Hom chom 17359   Func cfunc 17949   ⟨,⟩_F cprf 18265

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-map 8832  df-ixp 8909  df-func 17953  df-prf 18269

This theorem is used by:  prf2  18296