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Theorem prf2fval 17446
 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)
Assertion
Ref Expression
prf2fval (𝜑 → (𝑋(2nd𝑃)𝑌) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))
Distinct variable groups:   𝐵,   ,𝐹   𝜑,   ,𝐺   ,𝑋   ,𝑌   ,𝐻
Allowed substitution hints:   𝐶()   𝐷()   𝑃()   𝐸()

Proof of Theorem prf2fval
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 17444 . . 3 (𝜑𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
72fvexi 6663 . . . . 5 𝐵 ∈ V
87mptex 6967 . . . 4 (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
97, 7mpoex 7764 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
108, 9op2ndd 7686 . . 3 (𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (2nd𝑃) = (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
116, 10syl 17 . 2 (𝜑 → (2nd𝑃) = (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
12 simprl 770 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
13 simprr 772 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
1412, 13oveq12d 7157 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
1512, 13oveq12d 7157 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑌))
1615fveq1d 6651 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥(2nd𝐹)𝑦)‘) = ((𝑋(2nd𝐹)𝑌)‘))
1712, 13oveq12d 7157 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥(2nd𝐺)𝑦) = (𝑋(2nd𝐺)𝑌))
1817fveq1d 6651 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥(2nd𝐺)𝑦)‘) = ((𝑋(2nd𝐺)𝑌)‘))
1916, 18opeq12d 4776 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ = ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩)
2014, 19mpteq12dv 5118 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))
21 prf1.x . 2 (𝜑𝑋𝐵)
22 prf2.y . 2 (𝜑𝑌𝐵)
23 ovex 7172 . . . 4 (𝑋𝐻𝑌) ∈ V
2423mptex 6967 . . 3 ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩) ∈ V
2524a1i 11 . 2 (𝜑 → ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩) ∈ V)
2611, 20, 21, 22, 25ovmpod 7285 1 (𝜑 → (𝑋(2nd𝑃)𝑌) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ⟨cop 4534   ↦ cmpt 5113  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141  1st c1st 7673  2nd c2nd 7674  Basecbs 16478  Hom chom 16571   Func cfunc 17119   ⟨,⟩F cprf 17416 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-ixp 8449  df-func 17123  df-prf 17420 This theorem is referenced by:  prf2  17447
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