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Theorem prf1 16887
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 16886 . . 3 (𝜑𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
7 fvex 6239 . . . . . 6 (Base‘𝐶) ∈ V
82, 7eqeltri 2726 . . . . 5 𝐵 ∈ V
98mptex 6527 . . . 4 (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
108, 8mpt2ex 7292 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
119, 10op1std 7220 . . 3 (𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (1st𝑃) = (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
126, 11syl 17 . 2 (𝜑 → (1st𝑃) = (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
13 simpr 476 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
1413fveq2d 6233 . . 3 ((𝜑𝑥 = 𝑋) → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑋))
1513fveq2d 6233 . . 3 ((𝜑𝑥 = 𝑋) → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑋))
1614, 15opeq12d 4441 . 2 ((𝜑𝑥 = 𝑋) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩)
17 prf1.x . 2 (𝜑𝑋𝐵)
18 opex 4962 . . 3 ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩ ∈ V
1918a1i 11 . 2 (𝜑 → ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩ ∈ V)
2012, 16, 17, 19fvmptd 6327 1 (𝜑 → ((1st𝑃)‘𝑋) = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cop 4216  cmpt 4762  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  Basecbs 15904  Hom chom 15999   Func cfunc 16561   ⟨,⟩F cprf 16858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-ixp 7951  df-func 16565  df-prf 16862
This theorem is referenced by:  prfcl  16890  uncf1  16923  uncf2  16924  yonedalem21  16960  yonedalem22  16965
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