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Theorem uncf1 17303
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 17301 . . . 4 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6534 . . 3 (𝜑 → (1st𝐹) = (1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 7024 . 2 (𝜑 → (𝑋(1st𝐹)𝑌) = (𝑋(1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))𝑌))
8 df-ov 7010 . . 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 2793 . . . . 5 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
10 uncf1.a . . . . 5 𝐴 = (Base‘𝐶)
11 uncf1.b . . . . 5 𝐵 = (Base‘𝐷)
129, 10, 11xpcbas 17245 . . . 4 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
13 eqid 2793 . . . . 5 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
14 eqid 2793 . . . . 5 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
15 funcrcl 16950 . . . . . . . . 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 2793 . . . . . . 7 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
199, 17, 2, 181stfcl 17264 . . . . . 6 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2019, 4cofucl 16975 . . . . 5 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
21 eqid 2793 . . . . . 6 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
229, 17, 2, 212ndfcl 17265 . . . . 5 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2313, 14, 20, 22prfcl 17270 . . . 4 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
24 eqid 2793 . . . . 5 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
25 eqid 2793 . . . . 5 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2624, 25, 2, 3evlfcl 17289 . . . 4 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
27 uncf1.x . . . . 5 (𝜑𝑋𝐴)
28 uncf1.y . . . . 5 (𝜑𝑌𝐵)
2927, 28opelxpd 5473 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3012, 23, 26, 29cofu1 16971 . . 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 2841 . 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 2793 . . . . . . 7 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
3313, 12, 32, 20, 22, 29prf1 17267 . . . . . 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 16971 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
359, 12, 32, 17, 2, 18, 291stf1 17259 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
36 op1stg 7548 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3727, 28, 36syl2anc 584 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3835, 37eqtrd 2829 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
3938fveq2d 6534 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
4034, 39eqtrd 2829 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
419, 12, 32, 17, 2, 21, 292ndf1 17262 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
42 op2ndg 7549 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4327, 28, 42syl2anc 584 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4441, 43eqtrd 2829 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
4540, 44opeq12d 4712 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4633, 45eqtrd 2829 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4746fveq2d 6534 . . . 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 7010 . . . 4 (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘(𝐷 evalF 𝐸))‘⟨((1st𝐺)‘𝑋), 𝑌⟩)
4947, 48syl6eqr 2847 . . 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 17047 . . . . . 6 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
51 relfunc 16949 . . . . . . 7 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
52 1st2ndbr 7588 . . . . . . 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 16953 . . . . 5 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
5554, 27ffvelrnd 6708 . . . 4 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
5624, 2, 3, 11, 55, 28evlf1 17287 . . 3 (𝜑 → (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
5749, 56eqtrd 2829 . 2 (𝜑 → ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
587, 31, 573eqtrd 2833 1 (𝜑 → (𝑋(1st𝐹)𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1520  wcel 2079  cop 4472   class class class wbr 4956   × cxp 5433  Rel wrel 5440  cfv 6217  (class class class)co 7007  1st c1st 7534  2nd c2nd 7535  ⟨“cs3 14028  Basecbs 16300  Hom chom 16393  Catccat 16752   Func cfunc 16941  func ccofu 16943   FuncCat cfuc 17029   ×c cxpc 17235   1stF c1stf 17236   2ndF c2ndf 17237   ⟨,⟩F cprf 17238   evalF cevlf 17276   uncurryF cuncf 17278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-fal 1533  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-oadd 7948  df-er 8130  df-map 8249  df-ixp 8301  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-card 9203  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-5 11540  df-6 11541  df-7 11542  df-8 11543  df-9 11544  df-n0 11735  df-z 11819  df-dec 11937  df-uz 12083  df-fz 12732  df-fzo 12873  df-hash 13529  df-word 13696  df-concat 13757  df-s1 13782  df-s2 14034  df-s3 14035  df-struct 16302  df-ndx 16303  df-slot 16304  df-base 16306  df-hom 16406  df-cco 16407  df-cat 16756  df-cid 16757  df-func 16945  df-cofu 16947  df-nat 17030  df-fuc 17031  df-xpc 17239  df-1stf 17240  df-2ndf 17241  df-prf 17242  df-evlf 17280  df-uncf 17282
This theorem is referenced by:  curfuncf  17305  uncfcurf  17306
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