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Theorem uncf1 18043
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 18041 . . . 4 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6823 . . 3 (𝜑 → (1st𝐹) = (1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 7346 . 2 (𝜑 → (𝑋(1st𝐹)𝑌) = (𝑋(1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))𝑌))
8 df-ov 7332 . . 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 2736 . . . . 5 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
10 uncf1.a . . . . 5 𝐴 = (Base‘𝐶)
11 uncf1.b . . . . 5 𝐵 = (Base‘𝐷)
129, 10, 11xpcbas 17984 . . . 4 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
13 eqid 2736 . . . . 5 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
14 eqid 2736 . . . . 5 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
15 funcrcl 17667 . . . . . . . . 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 495 . . . . . . 7 (𝜑𝐶 ∈ Cat)
18 eqid 2736 . . . . . . 7 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
199, 17, 2, 181stfcl 18003 . . . . . 6 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2019, 4cofucl 17692 . . . . 5 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
21 eqid 2736 . . . . . 6 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
229, 17, 2, 212ndfcl 18004 . . . . 5 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2313, 14, 20, 22prfcl 18009 . . . 4 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
24 eqid 2736 . . . . 5 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
25 eqid 2736 . . . . 5 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2624, 25, 2, 3evlfcl 18029 . . . 4 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
27 uncf1.x . . . . 5 (𝜑𝑋𝐴)
28 uncf1.y . . . . 5 (𝜑𝑌𝐵)
2927, 28opelxpd 5652 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3012, 23, 26, 29cofu1 17688 . . 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, 30eqtrid 2788 . 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 2736 . . . . . . 7 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
3313, 12, 32, 20, 22, 29prf1 18006 . . . . . 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 17688 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
359, 12, 32, 17, 2, 18, 291stf1 17998 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
36 op1stg 7903 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3727, 28, 36syl2anc 584 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3835, 37eqtrd 2776 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
3938fveq2d 6823 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
4034, 39eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
419, 12, 32, 17, 2, 21, 292ndf1 18001 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
42 op2ndg 7904 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4327, 28, 42syl2anc 584 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4441, 43eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
4540, 44opeq12d 4824 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4633, 45eqtrd 2776 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4746fveq2d 6823 . . . 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 7332 . . . 4 (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘(𝐷 evalF 𝐸))‘⟨((1st𝐺)‘𝑋), 𝑌⟩)
4947, 48eqtr4di 2794 . . 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 17766 . . . . . 6 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
51 relfunc 17666 . . . . . . 7 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
52 1st2ndbr 7943 . . . . . . 7 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
5351, 4, 52sylancr 587 . . . . . 6 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
5410, 50, 53funcf1 17670 . . . . 5 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
5554, 27ffvelcdmd 7012 . . . 4 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
5624, 2, 3, 11, 55, 28evlf1 18027 . . 3 (𝜑 → (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
5749, 56eqtrd 2776 . 2 (𝜑 → ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
587, 31, 573eqtrd 2780 1 (𝜑 → (𝑋(1st𝐹)𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  cop 4578   class class class wbr 5089   × cxp 5612  Rel wrel 5619  cfv 6473  (class class class)co 7329  1st c1st 7889  2nd c2nd 7890  ⟨“cs3 14646  Basecbs 17001  Hom chom 17062  Catccat 17462   Func cfunc 17658  func ccofu 17660   FuncCat cfuc 17747   ×c cxpc 17974   1stF c1stf 17975   2ndF c2ndf 17976   ⟨,⟩F cprf 17977   evalF cevlf 18016   uncurryF cuncf 18018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-er 8561  df-map 8680  df-ixp 8749  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-card 9788  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-3 12130  df-4 12131  df-5 12132  df-6 12133  df-7 12134  df-8 12135  df-9 12136  df-n0 12327  df-z 12413  df-dec 12531  df-uz 12676  df-fz 13333  df-fzo 13476  df-hash 14138  df-word 14310  df-concat 14366  df-s1 14392  df-s2 14652  df-s3 14653  df-struct 16937  df-slot 16972  df-ndx 16984  df-base 17002  df-hom 17075  df-cco 17076  df-cat 17466  df-cid 17467  df-func 17662  df-cofu 17664  df-nat 17748  df-fuc 17749  df-xpc 17978  df-1stf 17979  df-2ndf 17980  df-prf 17981  df-evlf 18020  df-uncf 18022
This theorem is referenced by:  curfuncf  18045  uncfcurf  18046
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