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Theorem fvmpt2curryd 7554
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
fvmpt2curryd.f 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
fvmpt2curryd.c (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
fvmpt2curryd.y (𝜑𝑌𝑊)
fvmpt2curryd.a (𝜑𝐴𝑋)
fvmpt2curryd.b (𝜑𝐵𝑌)
Assertion
Ref Expression
fvmpt2curryd (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 fvmpt2curryd
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvmpt2curryd.b . . 3 (𝜑𝐵𝑌)
2 csbcom 4125 . . . . 5 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝐵 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶
3 csbco 3672 . . . . . 6 𝐵 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝑎 / 𝑥𝐶
43csbeq2i 4124 . . . . 5 𝐴 / 𝑎𝐵 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝐵 / 𝑦𝑎 / 𝑥𝐶
5 csbcom 4125 . . . . . 6 𝐴 / 𝑎𝐵 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑎𝑎 / 𝑥𝐶
6 csbco 3672 . . . . . . 7 𝐴 / 𝑎𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶
76csbeq2i 4124 . . . . . 6 𝐵 / 𝑦𝐴 / 𝑎𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
85, 7eqtri 2770 . . . . 5 𝐴 / 𝑎𝐵 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
92, 4, 83eqtri 2774 . . . 4 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
10 fvmpt2curryd.a . . . . 5 (𝜑𝐴𝑋)
11 fvmpt2curryd.c . . . . 5 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
12 nfcsb1v 3678 . . . . . . . 8 𝑥𝐴 / 𝑥𝐶
1312nfel1 2905 . . . . . . 7 𝑥𝐴 / 𝑥𝐶𝑉
14 nfcsb1v 3678 . . . . . . . 8 𝑦𝐵 / 𝑦𝐴 / 𝑥𝐶
1514nfel1 2905 . . . . . . 7 𝑦𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉
16 csbeq1a 3671 . . . . . . . 8 (𝑥 = 𝐴𝐶 = 𝐴 / 𝑥𝐶)
1716eleq1d 2812 . . . . . . 7 (𝑥 = 𝐴 → (𝐶𝑉𝐴 / 𝑥𝐶𝑉))
18 csbeq1a 3671 . . . . . . . 8 (𝑦 = 𝐵𝐴 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)
1918eleq1d 2812 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 / 𝑥𝐶𝑉𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉))
2013, 15, 17, 19rspc2 3447 . . . . . 6 ((𝐴𝑋𝐵𝑌) → (∀𝑥𝑋𝑦𝑌 𝐶𝑉𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉))
2120imp 444 . . . . 5 (((𝐴𝑋𝐵𝑌) ∧ ∀𝑥𝑋𝑦𝑌 𝐶𝑉) → 𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉)
2210, 1, 11, 21syl21anc 1462 . . . 4 (𝜑𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉)
239, 22syl5eqel 2831 . . 3 (𝜑𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉)
24 eqid 2748 . . . 4 (𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶) = (𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
2524fvmpts 6435 . . 3 ((𝐵𝑌𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉) → ((𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)‘𝐵) = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
261, 23, 25syl2anc 696 . 2 (𝜑 → ((𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)‘𝐵) = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
27 fvmpt2curryd.f . . . . 5 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
28 nfcv 2890 . . . . . 6 𝑎𝐶
29 nfcv 2890 . . . . . 6 𝑏𝐶
30 nfcv 2890 . . . . . . 7 𝑥𝑏
31 nfcsb1v 3678 . . . . . . 7 𝑥𝑎 / 𝑥𝐶
3230, 31nfcsb 3680 . . . . . 6 𝑥𝑏 / 𝑦𝑎 / 𝑥𝐶
33 nfcsb1v 3678 . . . . . 6 𝑦𝑏 / 𝑦𝑎 / 𝑥𝐶
34 csbeq1a 3671 . . . . . . 7 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
35 csbeq1a 3671 . . . . . . 7 (𝑦 = 𝑏𝑎 / 𝑥𝐶 = 𝑏 / 𝑦𝑎 / 𝑥𝐶)
3634, 35sylan9eq 2802 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐶 = 𝑏 / 𝑦𝑎 / 𝑥𝐶)
3728, 29, 32, 33, 36cbvmpt2 6887 . . . . 5 (𝑥𝑋, 𝑦𝑌𝐶) = (𝑎𝑋, 𝑏𝑌𝑏 / 𝑦𝑎 / 𝑥𝐶)
3827, 37eqtri 2770 . . . 4 𝐹 = (𝑎𝑋, 𝑏𝑌𝑏 / 𝑦𝑎 / 𝑥𝐶)
3931nfel1 2905 . . . . . . 7 𝑥𝑎 / 𝑥𝐶𝑉
4033nfel1 2905 . . . . . . 7 𝑦𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉
4134eleq1d 2812 . . . . . . 7 (𝑥 = 𝑎 → (𝐶𝑉𝑎 / 𝑥𝐶𝑉))
4235eleq1d 2812 . . . . . . 7 (𝑦 = 𝑏 → (𝑎 / 𝑥𝐶𝑉𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉))
4339, 40, 41, 42rspc2 3447 . . . . . 6 ((𝑎𝑋𝑏𝑌) → (∀𝑥𝑋𝑦𝑌 𝐶𝑉𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉))
4411, 43mpan9 487 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑌)) → 𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉)
4544ralrimivva 3097 . . . 4 (𝜑 → ∀𝑎𝑋𝑏𝑌 𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉)
46 ne0i 4052 . . . . 5 (𝐵𝑌𝑌 ≠ ∅)
471, 46syl 17 . . . 4 (𝜑𝑌 ≠ ∅)
48 fvmpt2curryd.y . . . 4 (𝜑𝑌𝑊)
4938, 45, 47, 48, 10mpt2curryvald 7553 . . 3 (𝜑 → (curry 𝐹𝐴) = (𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶))
5049fveq1d 6342 . 2 (𝜑 → ((curry 𝐹𝐴)‘𝐵) = ((𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)‘𝐵))
5127a1i 11 . . 3 (𝜑𝐹 = (𝑥𝑋, 𝑦𝑌𝐶))
52 csbco 3672 . . . . . . . 8 𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑦 / 𝑦𝑎 / 𝑥𝐶
53 csbid 3670 . . . . . . . 8 𝑦 / 𝑦𝑎 / 𝑥𝐶 = 𝑎 / 𝑥𝐶
5452, 53eqtr2i 2771 . . . . . . 7 𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶
5554a1i 11 . . . . . 6 (𝜑𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶)
5655csbeq2dv 4123 . . . . 5 (𝜑𝑥 / 𝑎𝑎 / 𝑥𝐶 = 𝑥 / 𝑎𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶)
57 csbco 3672 . . . . . 6 𝑥 / 𝑎𝑎 / 𝑥𝐶 = 𝑥 / 𝑥𝐶
58 csbid 3670 . . . . . 6 𝑥 / 𝑥𝐶 = 𝐶
5957, 58eqtri 2770 . . . . 5 𝑥 / 𝑎𝑎 / 𝑥𝐶 = 𝐶
60 csbcom 4125 . . . . 5 𝑥 / 𝑎𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
6156, 59, 603eqtr3g 2805 . . . 4 (𝜑𝐶 = 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
62 csbeq1 3665 . . . . . . 7 (𝑥 = 𝐴𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6362adantr 472 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6463csbeq2dv 4123 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
65 csbeq1 3665 . . . . . 6 (𝑦 = 𝐵𝑦 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6665adantl 473 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6764, 66eqtrd 2782 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6861, 67sylan9eq 2802 . . 3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
69 eqidd 2749 . . 3 ((𝜑𝑥 = 𝐴) → 𝑌 = 𝑌)
70 nfv 1980 . . 3 𝑥𝜑
71 nfv 1980 . . 3 𝑦𝜑
72 nfcv 2890 . . 3 𝑦𝐴
73 nfcv 2890 . . 3 𝑥𝐵
74 nfcv 2890 . . . . 5 𝑥𝐴
7574, 32nfcsb 3680 . . . 4 𝑥𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
7673, 75nfcsb 3680 . . 3 𝑥𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
779, 14nfcxfr 2888 . . 3 𝑦𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
7851, 68, 69, 10, 1, 23, 70, 71, 72, 73, 76, 77ovmpt2dxf 6939 . 2 (𝜑 → (𝐴𝐹𝐵) = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
7926, 50, 783eqtr4d 2792 1 (𝜑 → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  wne 2920  wral 3038  csb 3662  c0 4046  cmpt 4869  cfv 6037  (class class class)co 6801  cmpt2 6803  curry ccur 7548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-cur 7550
This theorem is referenced by:  pmatcollpw3lem  20761  logbfval  24698
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