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Theorem uncf1 17498
 Description: Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
uncfval.g 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
uncfval.c (𝜑𝐷 ∈ Cat)
uncfval.d (𝜑𝐸 ∈ Cat)
uncfval.f (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
uncf1.a 𝐴 = (Base‘𝐶)
uncf1.b 𝐵 = (Base‘𝐷)
uncf1.x (𝜑𝑋𝐴)
uncf1.y (𝜑𝑌𝐵)
Assertion
Ref Expression
uncf1 (𝜑 → (𝑋(1st𝐹)𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))

Proof of Theorem uncf1
StepHypRef Expression
1 uncfval.g . . . . 5 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2 uncfval.c . . . . 5 (𝜑𝐷 ∈ Cat)
3 uncfval.d . . . . 5 (𝜑𝐸 ∈ Cat)
4 uncfval.f . . . . 5 (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
51, 2, 3, 4uncfval 17496 . . . 4 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6659 . . 3 (𝜑 → (1st𝐹) = (1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 7162 . 2 (𝜑 → (𝑋(1st𝐹)𝑌) = (𝑋(1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))𝑌))
8 df-ov 7148 . . 3 (𝑋(1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))𝑌) = ((1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))‘⟨𝑋, 𝑌⟩)
9 eqid 2798 . . . . 5 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
10 uncf1.a . . . . 5 𝐴 = (Base‘𝐶)
11 uncf1.b . . . . 5 𝐵 = (Base‘𝐷)
129, 10, 11xpcbas 17440 . . . 4 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
13 eqid 2798 . . . . 5 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
14 eqid 2798 . . . . 5 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
15 funcrcl 17145 . . . . . . . . 9 (𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)) → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
164, 15syl 17 . . . . . . . 8 (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
1716simpld 498 . . . . . . 7 (𝜑𝐶 ∈ Cat)
18 eqid 2798 . . . . . . 7 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
199, 17, 2, 181stfcl 17459 . . . . . 6 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2019, 4cofucl 17170 . . . . 5 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
21 eqid 2798 . . . . . 6 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
229, 17, 2, 212ndfcl 17460 . . . . 5 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2313, 14, 20, 22prfcl 17465 . . . 4 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
24 eqid 2798 . . . . 5 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
25 eqid 2798 . . . . 5 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2624, 25, 2, 3evlfcl 17484 . . . 4 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
27 uncf1.x . . . . 5 (𝜑𝑋𝐴)
28 uncf1.y . . . . 5 (𝜑𝑌𝐵)
2927, 28opelxpd 5561 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3012, 23, 26, 29cofu1 17166 . . 3 (𝜑 → ((1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))‘⟨𝑋, 𝑌⟩) = ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)))
318, 30syl5eq 2845 . 2 (𝜑 → (𝑋(1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))𝑌) = ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)))
32 eqid 2798 . . . . . . 7 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
3313, 12, 32, 20, 22, 29prf1 17462 . . . . . 6 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩)
3412, 19, 4, 29cofu1 17166 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
359, 12, 32, 17, 2, 18, 291stf1 17454 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
36 op1stg 7696 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3727, 28, 36syl2anc 587 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3835, 37eqtrd 2833 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
3938fveq2d 6659 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
4034, 39eqtrd 2833 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
419, 12, 32, 17, 2, 21, 292ndf1 17457 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
42 op2ndg 7697 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4327, 28, 42syl2anc 587 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4441, 43eqtrd 2833 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
4540, 44opeq12d 4777 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4633, 45eqtrd 2833 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4746fveq2d 6659 . . . 4 (𝜑 → ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)) = ((1st ‘(𝐷 evalF 𝐸))‘⟨((1st𝐺)‘𝑋), 𝑌⟩))
48 df-ov 7148 . . . 4 (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘(𝐷 evalF 𝐸))‘⟨((1st𝐺)‘𝑋), 𝑌⟩)
4947, 48eqtr4di 2851 . . 3 (𝜑 → ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)) = (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌))
5025fucbas 17242 . . . . . 6 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
51 relfunc 17144 . . . . . . 7 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
52 1st2ndbr 7736 . . . . . . 7 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
5351, 4, 52sylancr 590 . . . . . 6 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
5410, 50, 53funcf1 17148 . . . . 5 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
5554, 27ffvelrnd 6839 . . . 4 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
5624, 2, 3, 11, 55, 28evlf1 17482 . . 3 (𝜑 → (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
5749, 56eqtrd 2833 . 2 (𝜑 → ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
587, 31, 573eqtrd 2837 1 (𝜑 → (𝑋(1st𝐹)𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4534   class class class wbr 5034   × cxp 5521  Rel wrel 5528  ‘cfv 6332  (class class class)co 7145  1st c1st 7682  2nd c2nd 7683  ⟨“cs3 14215  Basecbs 16495  Hom chom 16588  Catccat 16947   Func cfunc 17136   ∘func ccofu 17138   FuncCat cfuc 17224   ×c cxpc 17430   1stF c1stf 17431   2ndF c2ndf 17432   ⟨,⟩F cprf 17433   evalF cevlf 17471   uncurryF cuncf 17473 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 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  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-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  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-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-map 8409  df-ixp 8463  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-3 11707  df-4 11708  df-5 11709  df-6 11710  df-7 11711  df-8 11712  df-9 11713  df-n0 11904  df-z 11990  df-dec 12107  df-uz 12252  df-fz 12906  df-fzo 13049  df-hash 13707  df-word 13878  df-concat 13934  df-s1 13961  df-s2 14221  df-s3 14222  df-struct 16497  df-ndx 16498  df-slot 16499  df-base 16501  df-hom 16601  df-cco 16602  df-cat 16951  df-cid 16952  df-func 17140  df-cofu 17142  df-nat 17225  df-fuc 17226  df-xpc 17434  df-1stf 17435  df-2ndf 17436  df-prf 17437  df-evlf 17475  df-uncf 17477 This theorem is referenced by:  curfuncf  17500  uncfcurf  17501
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