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Theorem uncf1 17077
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 17075 . . . 4 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6334 . . 3 (𝜑 → (1st𝐹) = (1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 6808 . 2 (𝜑 → (𝑋(1st𝐹)𝑌) = (𝑋(1st ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))𝑌))
8 df-ov 6794 . . 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 2771 . . . . 5 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
10 uncf1.a . . . . 5 𝐴 = (Base‘𝐶)
11 uncf1.b . . . . 5 𝐵 = (Base‘𝐷)
129, 10, 11xpcbas 17019 . . . 4 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
13 eqid 2771 . . . . 5 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
14 eqid 2771 . . . . 5 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
15 funcrcl 16723 . . . . . . . . 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 482 . . . . . . 7 (𝜑𝐶 ∈ Cat)
18 eqid 2771 . . . . . . 7 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
199, 17, 2, 181stfcl 17038 . . . . . 6 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2019, 4cofucl 16748 . . . . 5 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
21 eqid 2771 . . . . . 6 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
229, 17, 2, 212ndfcl 17039 . . . . 5 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2313, 14, 20, 22prfcl 17044 . . . 4 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
24 eqid 2771 . . . . 5 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
25 eqid 2771 . . . . 5 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2624, 25, 2, 3evlfcl 17063 . . . 4 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
27 uncf1.x . . . . 5 (𝜑𝑋𝐴)
28 uncf1.y . . . . 5 (𝜑𝑌𝐵)
29 opelxpi 5286 . . . . 5 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3027, 28, 29syl2anc 573 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3112, 23, 26, 30cofu1 16744 . . 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 2817 . 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 2771 . . . . . . 7 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
3413, 12, 33, 20, 22, 30prf1 17041 . . . . . 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 16744 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
369, 12, 33, 17, 2, 18, 301stf1 17033 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
37 op1stg 7325 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3827, 28, 37syl2anc 573 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3936, 38eqtrd 2805 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
4039fveq2d 6334 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
4135, 40eqtrd 2805 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
429, 12, 33, 17, 2, 21, 302ndf1 17036 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
43 op2ndg 7326 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4427, 28, 43syl2anc 573 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
4542, 44eqtrd 2805 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
4641, 45opeq12d 4547 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4734, 46eqtrd 2805 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
4847fveq2d 6334 . . . 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 6794 . . . 4 (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘(𝐷 evalF 𝐸))‘⟨((1st𝐺)‘𝑋), 𝑌⟩)
5048, 49syl6eqr 2823 . . 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 16820 . . . . . 6 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
52 relfunc 16722 . . . . . . 7 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
53 1st2ndbr 7364 . . . . . . 7 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
5452, 4, 53sylancr 575 . . . . . 6 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
5510, 51, 54funcf1 16726 . . . . 5 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
5655, 27ffvelrnd 6501 . . . 4 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
5724, 2, 3, 11, 56, 28evlf1 17061 . . 3 (𝜑 → (((1st𝐺)‘𝑋)(1st ‘(𝐷 evalF 𝐸))𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
5850, 57eqtrd 2805 . 2 (𝜑 → ((1st ‘(𝐷 evalF 𝐸))‘((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
597, 32, 583eqtrd 2809 1 (𝜑 → (𝑋(1st𝐹)𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cop 4322   class class class wbr 4786   × cxp 5247  Rel wrel 5254  cfv 6029  (class class class)co 6791  1st c1st 7311  2nd c2nd 7312  ⟨“cs3 13789  Basecbs 16057  Hom chom 16153  Catccat 16525   Func cfunc 16714  func ccofu 16716   FuncCat cfuc 16802   ×c cxpc 17009   1stF c1stf 17010   2ndF c2ndf 17011   ⟨,⟩F cprf 17012   evalF cevlf 17050   uncurryF cuncf 17052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-ixp 8061  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-card 8963  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-2 11279  df-3 11280  df-4 11281  df-5 11282  df-6 11283  df-7 11284  df-8 11285  df-9 11286  df-n0 11493  df-z 11578  df-dec 11694  df-uz 11887  df-fz 12527  df-fzo 12667  df-hash 13315  df-word 13488  df-concat 13490  df-s1 13491  df-s2 13795  df-s3 13796  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-hom 16167  df-cco 16168  df-cat 16529  df-cid 16530  df-func 16718  df-cofu 16720  df-nat 16803  df-fuc 16804  df-xpc 17013  df-1stf 17014  df-2ndf 17015  df-prf 17016  df-evlf 17054  df-uncf 17056
This theorem is referenced by:  curfuncf  17079  uncfcurf  17080
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