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Theorem prf2fval 16963
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 16961 . . 3 (𝜑𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
7 fvex 6314 . . . . . 6 (Base‘𝐶) ∈ V
82, 7eqeltri 2799 . . . . 5 𝐵 ∈ V
98mptex 6602 . . . 4 (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
108, 8mpt2ex 7367 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
119, 10op2ndd 7296 . . 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)‘)⟩)))
126, 11syl 17 . 2 (𝜑 → (2nd𝑃) = (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
13 simprl 811 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
14 simprr 813 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
1513, 14oveq12d 6783 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
1613, 14oveq12d 6783 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑌))
1716fveq1d 6306 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥(2nd𝐹)𝑦)‘) = ((𝑋(2nd𝐹)𝑌)‘))
1813, 14oveq12d 6783 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥(2nd𝐺)𝑦) = (𝑋(2nd𝐺)𝑌))
1918fveq1d 6306 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥(2nd𝐺)𝑦)‘) = ((𝑋(2nd𝐺)𝑌)‘))
2017, 19opeq12d 4517 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ = ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩)
2115, 20mpteq12dv 4841 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))
22 prf1.x . 2 (𝜑𝑋𝐵)
23 prf2.y . 2 (𝜑𝑌𝐵)
24 ovex 6793 . . . 4 (𝑋𝐻𝑌) ∈ V
2524mptex 6602 . . 3 ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩) ∈ V
2625a1i 11 . 2 (𝜑 → ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩) ∈ V)
2712, 21, 22, 23, 26ovmpt2d 6905 1 (𝜑 → (𝑋(2nd𝑃)𝑌) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  Vcvv 3304  cop 4291  cmpt 4837  cfv 6001  (class class class)co 6765  cmpt2 6767  1st c1st 7283  2nd c2nd 7284  Basecbs 15980  Hom chom 16075   Func cfunc 16636   ⟨,⟩F cprf 16933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-map 7976  df-ixp 8026  df-func 16640  df-prf 16937
This theorem is referenced by:  prf2  16964
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