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Theorem uncf1 16797
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 16795 . . . 4 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6152 . . 3 (𝜑 → (1st𝐹) = (1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 6621 . 2 (𝜑 → (𝑋(1st𝐹)𝑌) = (𝑋(1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))𝑌))
8 df-ov 6607 . . 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 2621 . . . . 5 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
10 uncf1.a . . . . 5 𝐴 = (Base‘𝐶)
11 uncf1.b . . . . 5 𝐵 = (Base‘𝐷)
129, 10, 11xpcbas 16739 . . . 4 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
13 eqid 2621 . . . . 5 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
14 eqid 2621 . . . . 5 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
15 funcrcl 16444 . . . . . . . . 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 475 . . . . . . 7 (𝜑𝐶 ∈ Cat)
18 eqid 2621 . . . . . . 7 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
199, 17, 2, 181stfcl 16758 . . . . . 6 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2019, 4cofucl 16469 . . . . 5 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
21 eqid 2621 . . . . . 6 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
229, 17, 2, 212ndfcl 16759 . . . . 5 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2313, 14, 20, 22prfcl 16764 . . . 4 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
24 eqid 2621 . . . . 5 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
25 eqid 2621 . . . . 5 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2624, 25, 2, 3evlfcl 16783 . . . 4 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
27 uncf1.x . . . . 5 (𝜑𝑋𝐴)
28 uncf1.y . . . . 5 (𝜑𝑌𝐵)
29 opelxpi 5108 . . . . 5 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3027, 28, 29syl2anc 692 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3112, 23, 26, 30cofu1 16465 . . 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⟩)))
328, 31syl5eq 2667 . 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⟩)))
33 eqid 2621 . . . . . . 7 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
3413, 12, 33, 20, 22, 30prf1 16761 . . . . . 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⟩)⟩)
3512, 19, 4, 30cofu1 16465 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
369, 12, 33, 17, 2, 18, 301stf1 16753 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
37 op1stg 7125 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3827, 28, 37syl2anc 692 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3936, 38eqtrd 2655 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
4039fveq2d 6152 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
4135, 40eqtrd 2655 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
429, 12, 33, 17, 2, 21, 302ndf1 16756 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
43 op2ndg 7126 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4427, 28, 43syl2anc 692 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4542, 44eqtrd 2655 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
4641, 45opeq12d 4378 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4734, 46eqtrd 2655 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4847fveq2d 6152 . . . 4 (𝜑 → ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)) = ((1st ‘(𝐷 evalF 𝐸))‘⟨((1st𝐺)‘𝑋), 𝑌⟩))
49 df-ov 6607 . . . 4 (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘(𝐷 evalF 𝐸))‘⟨((1st𝐺)‘𝑋), 𝑌⟩)
5048, 49syl6eqr 2673 . . 3 (𝜑 → ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)) = (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌))
5125fucbas 16541 . . . . . 6 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
52 relfunc 16443 . . . . . . 7 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
53 1st2ndbr 7162 . . . . . . 7 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
5452, 4, 53sylancr 694 . . . . . 6 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
5510, 51, 54funcf1 16447 . . . . 5 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
5655, 27ffvelrnd 6316 . . . 4 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
5724, 2, 3, 11, 56, 28evlf1 16781 . . 3 (𝜑 → (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
5850, 57eqtrd 2655 . 2 (𝜑 → ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
597, 32, 583eqtrd 2659 1 (𝜑 → (𝑋(1st𝐹)𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cop 4154   class class class wbr 4613   × cxp 5072  Rel wrel 5079  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  ⟨“cs3 13524  Basecbs 15781  Hom chom 15873  Catccat 16246   Func cfunc 16435  func ccofu 16437   FuncCat cfuc 16523   ×c cxpc 16729   1stF c1stf 16730   2ndF c2ndf 16731   ⟨,⟩F cprf 16732   evalF cevlf 16770   uncurryF cuncf 16772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-s2 13530  df-s3 13531  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-hom 15887  df-cco 15888  df-cat 16250  df-cid 16251  df-func 16439  df-cofu 16441  df-nat 16524  df-fuc 16525  df-xpc 16733  df-1stf 16734  df-2ndf 16735  df-prf 16736  df-evlf 16774  df-uncf 16776
This theorem is referenced by:  curfuncf  16799  uncfcurf  16800
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