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Theorem prf2 18186
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 18185 . 2 (𝜑 → (𝑋(2nd𝑃)𝑌) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))
9 simpr 484 . . . 4 ((𝜑 = 𝐾) → = 𝐾)
109fveq2d 6895 . . 3 ((𝜑 = 𝐾) → ((𝑋(2nd𝐹)𝑌)‘) = ((𝑋(2nd𝐹)𝑌)‘𝐾))
119fveq2d 6895 . . 3 ((𝜑 = 𝐾) → ((𝑋(2nd𝐺)𝑌)‘) = ((𝑋(2nd𝐺)𝑌)‘𝐾))
1210, 11opeq12d 4877 . 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 5460 . . 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 7006 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 1534  wcel 2099  Vcvv 3470  cop 4630  cfv 6542  (class class class)co 7414  2nd c2nd 7986  Basecbs 17173  Hom chom 17237   Func cfunc 17833   ⟨,⟩F cprf 18155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8840  df-ixp 8910  df-func 17837  df-prf 18159
This theorem is referenced by:  prfcl  18187  prf1st  18188  prf2nd  18189  uncf2  18222  yonedalem22  18263
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