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Theorem prf1 17298
 Description: Value of the pairing functor on objects. (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 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
prfval.d (𝜑𝐺 ∈ (𝐶 Func 𝐸))
prf1.x (𝜑𝑋𝐵)
Assertion
Ref Expression
prf1 (𝜑 → ((1st𝑃)‘𝑋) = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩)

Proof of Theorem prf1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef 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 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
5 prfval.d . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
61, 2, 3, 4, 5prfval 17297 . . 3 (𝜑𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
72fvexi 6507 . . . . 5 𝐵 ∈ V
87mptex 6806 . . . 4 (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
97, 7mpoex 7578 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
108, 9op1std 7504 . . 3 (𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (1st𝑃) = (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
116, 10syl 17 . 2 (𝜑 → (1st𝑃) = (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
12 simpr 477 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
1312fveq2d 6497 . . 3 ((𝜑𝑥 = 𝑋) → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑋))
1412fveq2d 6497 . . 3 ((𝜑𝑥 = 𝑋) → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑋))
1513, 14opeq12d 4679 . 2 ((𝜑𝑥 = 𝑋) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩)
16 prf1.x . 2 (𝜑𝑋𝐵)
17 opex 5206 . . 3 ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩ ∈ V
1817a1i 11 . 2 (𝜑 → ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩ ∈ V)
1911, 15, 16, 18fvmptd 6595 1 (𝜑 → ((1st𝑃)‘𝑋) = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2048  Vcvv 3409  ⟨cop 4441   ↦ cmpt 5002  ‘cfv 6182  (class class class)co 6970   ∈ cmpo 6972  1st c1st 7492  2nd c2nd 7493  Basecbs 16329  Hom chom 16422   Func cfunc 16972   ⟨,⟩F cprf 17269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-map 8200  df-ixp 8252  df-func 16976  df-prf 17273 This theorem is referenced by:  prfcl  17301  uncf1  17334  uncf2  17335  yonedalem21  17371  yonedalem22  17376
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