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Theorem prf2 17930
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 17929 . 2 (𝜑 → (𝑋(2nd𝑃)𝑌) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))
9 simpr 485 . . . 4 ((𝜑 = 𝐾) → = 𝐾)
109fveq2d 6775 . . 3 ((𝜑 = 𝐾) → ((𝑋(2nd𝐹)𝑌)‘) = ((𝑋(2nd𝐹)𝑌)‘𝐾))
119fveq2d 6775 . . 3 ((𝜑 = 𝐾) → ((𝑋(2nd𝐺)𝑌)‘) = ((𝑋(2nd𝐺)𝑌)‘𝐾))
1210, 11opeq12d 4818 . 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 5383 . . 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 6879 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 396   = wceq 1542  wcel 2110  Vcvv 3431  cop 4573  cfv 6432  (class class class)co 7272  2nd c2nd 7824  Basecbs 16923  Hom chom 16984   Func cfunc 17580   ⟨,⟩F cprf 17899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7275  df-oprab 7276  df-mpo 7277  df-1st 7825  df-2nd 7826  df-map 8609  df-ixp 8678  df-func 17584  df-prf 17903
This theorem is referenced by:  prfcl  17931  prf1st  17932  prf2nd  17933  uncf2  17966  yonedalem22  18007
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