MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prf2 Structured version   Visualization version   GIF version

Theorem prf2 17444
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 17443 . 2 (𝜑 → (𝑋(2nd𝑃)𝑌) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))
9 simpr 488 . . . 4 ((𝜑 = 𝐾) → = 𝐾)
109fveq2d 6649 . . 3 ((𝜑 = 𝐾) → ((𝑋(2nd𝐹)𝑌)‘) = ((𝑋(2nd𝐹)𝑌)‘𝐾))
119fveq2d 6649 . . 3 ((𝜑 = 𝐾) → ((𝑋(2nd𝐺)𝑌)‘) = ((𝑋(2nd𝐺)𝑌)‘𝐾))
1210, 11opeq12d 4773 . 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 5321 . . 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 6752 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 399   = wceq 1538  wcel 2111  Vcvv 3441  cop 4531  cfv 6324  (class class class)co 7135  2nd c2nd 7670  Basecbs 16475  Hom chom 16568   Func cfunc 17116   ⟨,⟩F cprf 17413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-ixp 8445  df-func 17120  df-prf 17417
This theorem is referenced by:  prfcl  17445  prf1st  17446  prf2nd  17447  uncf2  17479  yonedalem22  17520
  Copyright terms: Public domain W3C validator