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Theorem prf2 18247
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 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
prfval.d (𝜑𝐺 ∈ (𝐶 Func 𝐸))
prf1.x (𝜑𝑋𝐵)
prf2.y (𝜑𝑌𝐵)
prf2.k (𝜑𝐾 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
prf2 (𝜑 → ((𝑋(2nd𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd𝐹)𝑌)‘𝐾), ((𝑋(2nd𝐺)𝑌)‘𝐾)⟩)

Proof of Theorem prf2
Dummy variable is distinct from all other variables.
StepHypRef 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 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
5 prfval.d . . 3 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
6 prf1.x . . 3 (𝜑𝑋𝐵)
7 prf2.y . . 3 (𝜑𝑌𝐵)
81, 2, 3, 4, 5, 6, 7prf2fval 18246 . 2 (𝜑 → (𝑋(2nd𝑃)𝑌) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))
9 simpr 484 . . . 4 ((𝜑 = 𝐾) → = 𝐾)
109fveq2d 6910 . . 3 ((𝜑 = 𝐾) → ((𝑋(2nd𝐹)𝑌)‘) = ((𝑋(2nd𝐹)𝑌)‘𝐾))
119fveq2d 6910 . . 3 ((𝜑 = 𝐾) → ((𝑋(2nd𝐺)𝑌)‘) = ((𝑋(2nd𝐺)𝑌)‘𝐾))
1210, 11opeq12d 4881 . 2 ((𝜑 = 𝐾) → ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩ = ⟨((𝑋(2nd𝐹)𝑌)‘𝐾), ((𝑋(2nd𝐺)𝑌)‘𝐾)⟩)
13 prf2.k . 2 (𝜑𝐾 ∈ (𝑋𝐻𝑌))
14 opex 5469 . . 3 ⟨((𝑋(2nd𝐹)𝑌)‘𝐾), ((𝑋(2nd𝐺)𝑌)‘𝐾)⟩ ∈ V
1514a1i 11 . 2 (𝜑 → ⟨((𝑋(2nd𝐹)𝑌)‘𝐾), ((𝑋(2nd𝐺)𝑌)‘𝐾)⟩ ∈ V)
168, 12, 13, 15fvmptd 7023 1 (𝜑 → ((𝑋(2nd𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd𝐹)𝑌)‘𝐾), ((𝑋(2nd𝐺)𝑌)‘𝐾)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cfv 6561  (class class class)co 7431  2nd c2nd 8013  Basecbs 17247  Hom chom 17308   Func cfunc 17899   ⟨,⟩F cprf 18216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ixp 8938  df-func 17903  df-prf 18220
This theorem is referenced by:  prfcl  18248  prf1st  18249  prf2nd  18250  uncf2  18282  yonedalem22  18323
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