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Theorem fvmpocurryd 7931
Description: The value of the value of a curried operation given in maps-to notation is the operation value of the original operation. (Contributed by AV, 27-Oct-2019.)
Hypotheses
Ref Expression
fvmpocurryd.f 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
fvmpocurryd.c (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
fvmpocurryd.y (𝜑𝑌𝑊)
fvmpocurryd.a (𝜑𝐴𝑋)
fvmpocurryd.b (𝜑𝐵𝑌)
Assertion
Ref Expression
fvmpocurryd (𝜑 → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem fvmpocurryd
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvmpocurryd.b . . 3 (𝜑𝐵𝑌)
2 csbcom 4372 . . . . 5 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝐵 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶
3 csbcow 3901 . . . . . 6 𝐵 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝑎 / 𝑥𝐶
43csbeq2i 3894 . . . . 5 𝐴 / 𝑎𝐵 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝐵 / 𝑦𝑎 / 𝑥𝐶
5 csbcom 4372 . . . . . 6 𝐴 / 𝑎𝐵 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑎𝑎 / 𝑥𝐶
6 csbcow 3901 . . . . . . 7 𝐴 / 𝑎𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶
76csbeq2i 3894 . . . . . 6 𝐵 / 𝑦𝐴 / 𝑎𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
85, 7eqtri 2848 . . . . 5 𝐴 / 𝑎𝐵 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
92, 4, 83eqtri 2852 . . . 4 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
10 fvmpocurryd.a . . . . 5 (𝜑𝐴𝑋)
11 fvmpocurryd.c . . . . 5 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
12 nfcsb1v 3910 . . . . . . . 8 𝑥𝐴 / 𝑥𝐶
1312nfel1 2998 . . . . . . 7 𝑥𝐴 / 𝑥𝐶𝑉
14 nfcsb1v 3910 . . . . . . . 8 𝑦𝐵 / 𝑦𝐴 / 𝑥𝐶
1514nfel1 2998 . . . . . . 7 𝑦𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉
16 csbeq1a 3900 . . . . . . . 8 (𝑥 = 𝐴𝐶 = 𝐴 / 𝑥𝐶)
1716eleq1d 2901 . . . . . . 7 (𝑥 = 𝐴 → (𝐶𝑉𝐴 / 𝑥𝐶𝑉))
18 csbeq1a 3900 . . . . . . . 8 (𝑦 = 𝐵𝐴 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)
1918eleq1d 2901 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 / 𝑥𝐶𝑉𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉))
2013, 15, 17, 19rspc2 3634 . . . . . 6 ((𝐴𝑋𝐵𝑌) → (∀𝑥𝑋𝑦𝑌 𝐶𝑉𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉))
2120imp 407 . . . . 5 (((𝐴𝑋𝐵𝑌) ∧ ∀𝑥𝑋𝑦𝑌 𝐶𝑉) → 𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉)
2210, 1, 11, 21syl21anc 835 . . . 4 (𝜑𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉)
239, 22eqeltrid 2921 . . 3 (𝜑𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉)
24 eqid 2824 . . . 4 (𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶) = (𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
2524fvmpts 6767 . . 3 ((𝐵𝑌𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉) → ((𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)‘𝐵) = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
261, 23, 25syl2anc 584 . 2 (𝜑 → ((𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)‘𝐵) = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
27 fvmpocurryd.f . . . . 5 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
28 nfcv 2981 . . . . . 6 𝑎𝐶
29 nfcv 2981 . . . . . 6 𝑏𝐶
30 nfcv 2981 . . . . . . 7 𝑥𝑏
31 nfcsb1v 3910 . . . . . . 7 𝑥𝑎 / 𝑥𝐶
3230, 31nfcsbw 3912 . . . . . 6 𝑥𝑏 / 𝑦𝑎 / 𝑥𝐶
33 nfcsb1v 3910 . . . . . 6 𝑦𝑏 / 𝑦𝑎 / 𝑥𝐶
34 csbeq1a 3900 . . . . . . 7 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
35 csbeq1a 3900 . . . . . . 7 (𝑦 = 𝑏𝑎 / 𝑥𝐶 = 𝑏 / 𝑦𝑎 / 𝑥𝐶)
3634, 35sylan9eq 2880 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐶 = 𝑏 / 𝑦𝑎 / 𝑥𝐶)
3728, 29, 32, 33, 36cbvmpo 7241 . . . . 5 (𝑥𝑋, 𝑦𝑌𝐶) = (𝑎𝑋, 𝑏𝑌𝑏 / 𝑦𝑎 / 𝑥𝐶)
3827, 37eqtri 2848 . . . 4 𝐹 = (𝑎𝑋, 𝑏𝑌𝑏 / 𝑦𝑎 / 𝑥𝐶)
3931nfel1 2998 . . . . . . 7 𝑥𝑎 / 𝑥𝐶𝑉
4033nfel1 2998 . . . . . . 7 𝑦𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉
4134eleq1d 2901 . . . . . . 7 (𝑥 = 𝑎 → (𝐶𝑉𝑎 / 𝑥𝐶𝑉))
4235eleq1d 2901 . . . . . . 7 (𝑦 = 𝑏 → (𝑎 / 𝑥𝐶𝑉𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉))
4339, 40, 41, 42rspc2 3634 . . . . . 6 ((𝑎𝑋𝑏𝑌) → (∀𝑥𝑋𝑦𝑌 𝐶𝑉𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉))
4411, 43mpan9 507 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑌)) → 𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉)
4544ralrimivva 3195 . . . 4 (𝜑 → ∀𝑎𝑋𝑏𝑌 𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉)
461ne0d 4304 . . . 4 (𝜑𝑌 ≠ ∅)
47 fvmpocurryd.y . . . 4 (𝜑𝑌𝑊)
4838, 45, 46, 47, 10mpocurryvald 7930 . . 3 (𝜑 → (curry 𝐹𝐴) = (𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶))
4948fveq1d 6668 . 2 (𝜑 → ((curry 𝐹𝐴)‘𝐵) = ((𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)‘𝐵))
5027a1i 11 . . 3 (𝜑𝐹 = (𝑥𝑋, 𝑦𝑌𝐶))
51 csbcow 3901 . . . . . . . 8 𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑦 / 𝑦𝑎 / 𝑥𝐶
52 csbid 3899 . . . . . . . 8 𝑦 / 𝑦𝑎 / 𝑥𝐶 = 𝑎 / 𝑥𝐶
5351, 52eqtr2i 2849 . . . . . . 7 𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶
5453a1i 11 . . . . . 6 (𝜑𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶)
5554csbeq2dv 3893 . . . . 5 (𝜑𝑥 / 𝑎𝑎 / 𝑥𝐶 = 𝑥 / 𝑎𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶)
56 csbcow 3901 . . . . . 6 𝑥 / 𝑎𝑎 / 𝑥𝐶 = 𝑥 / 𝑥𝐶
57 csbid 3899 . . . . . 6 𝑥 / 𝑥𝐶 = 𝐶
5856, 57eqtri 2848 . . . . 5 𝑥 / 𝑎𝑎 / 𝑥𝐶 = 𝐶
59 csbcom 4372 . . . . 5 𝑥 / 𝑎𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
6055, 58, 593eqtr3g 2883 . . . 4 (𝜑𝐶 = 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
61 csbeq1 3889 . . . . . . 7 (𝑥 = 𝐴𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6261adantr 481 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6362csbeq2dv 3893 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
64 csbeq1 3889 . . . . . 6 (𝑦 = 𝐵𝑦 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6564adantl 482 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6663, 65eqtrd 2860 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6760, 66sylan9eq 2880 . . 3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
68 eqidd 2825 . . 3 ((𝜑𝑥 = 𝐴) → 𝑌 = 𝑌)
69 nfv 1908 . . 3 𝑥𝜑
70 nfv 1908 . . 3 𝑦𝜑
71 nfcv 2981 . . 3 𝑦𝐴
72 nfcv 2981 . . 3 𝑥𝐵
73 nfcv 2981 . . . . 5 𝑥𝐴
7473, 32nfcsbw 3912 . . . 4 𝑥𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
7572, 74nfcsbw 3912 . . 3 𝑥𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
769, 14nfcxfr 2979 . . 3 𝑦𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
7750, 67, 68, 10, 1, 23, 69, 70, 71, 72, 75, 76ovmpodxf 7293 . 2 (𝜑 → (𝐴𝐹𝐵) = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
7826, 49, 773eqtr4d 2870 1 (𝜑 → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2106  wral 3142  csb 3886  cmpt 5142  cfv 6351  (class class class)co 7151  cmpo 7153  curry ccur 7925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-cur 7927
This theorem is referenced by:  pmatcollpw3lem  21307  logbfval  25281
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