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Theorem fvmpt2curryd 7604
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 4157 . . . . 5 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝐵 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶
3 csbco 3703 . . . . . 6 𝐵 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝑎 / 𝑥𝐶
43csbeq2i 4156 . . . . 5 𝐴 / 𝑎𝐵 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝐵 / 𝑦𝑎 / 𝑥𝐶
5 csbcom 4157 . . . . . 6 𝐴 / 𝑎𝐵 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑎𝑎 / 𝑥𝐶
6 csbco 3703 . . . . . . 7 𝐴 / 𝑎𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶
76csbeq2i 4156 . . . . . 6 𝐵 / 𝑦𝐴 / 𝑎𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
85, 7eqtri 2787 . . . . 5 𝐴 / 𝑎𝐵 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
92, 4, 83eqtri 2791 . . . 4 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
10 fvmpt2curryd.a . . . . 5 (𝜑𝐴𝑋)
11 fvmpt2curryd.c . . . . 5 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
12 nfcsb1v 3709 . . . . . . . 8 𝑥𝐴 / 𝑥𝐶
1312nfel1 2922 . . . . . . 7 𝑥𝐴 / 𝑥𝐶𝑉
14 nfcsb1v 3709 . . . . . . . 8 𝑦𝐵 / 𝑦𝐴 / 𝑥𝐶
1514nfel1 2922 . . . . . . 7 𝑦𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉
16 csbeq1a 3702 . . . . . . . 8 (𝑥 = 𝐴𝐶 = 𝐴 / 𝑥𝐶)
1716eleq1d 2829 . . . . . . 7 (𝑥 = 𝐴 → (𝐶𝑉𝐴 / 𝑥𝐶𝑉))
18 csbeq1a 3702 . . . . . . . 8 (𝑦 = 𝐵𝐴 / 𝑥𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)
1918eleq1d 2829 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 / 𝑥𝐶𝑉𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉))
2013, 15, 17, 19rspc2 3473 . . . . . 6 ((𝐴𝑋𝐵𝑌) → (∀𝑥𝑋𝑦𝑌 𝐶𝑉𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉))
2120imp 395 . . . . 5 (((𝐴𝑋𝐵𝑌) ∧ ∀𝑥𝑋𝑦𝑌 𝐶𝑉) → 𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉)
2210, 1, 11, 21syl21anc 866 . . . 4 (𝜑𝐵 / 𝑦𝐴 / 𝑥𝐶𝑉)
239, 22syl5eqel 2848 . . 3 (𝜑𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉)
24 eqid 2765 . . . 4 (𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶) = (𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
2524fvmpts 6478 . . 3 ((𝐵𝑌𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉) → ((𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)‘𝐵) = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
261, 23, 25syl2anc 579 . 2 (𝜑 → ((𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)‘𝐵) = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
27 fvmpt2curryd.f . . . . 5 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
28 nfcv 2907 . . . . . 6 𝑎𝐶
29 nfcv 2907 . . . . . 6 𝑏𝐶
30 nfcv 2907 . . . . . . 7 𝑥𝑏
31 nfcsb1v 3709 . . . . . . 7 𝑥𝑎 / 𝑥𝐶
3230, 31nfcsb 3711 . . . . . 6 𝑥𝑏 / 𝑦𝑎 / 𝑥𝐶
33 nfcsb1v 3709 . . . . . 6 𝑦𝑏 / 𝑦𝑎 / 𝑥𝐶
34 csbeq1a 3702 . . . . . . 7 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
35 csbeq1a 3702 . . . . . . 7 (𝑦 = 𝑏𝑎 / 𝑥𝐶 = 𝑏 / 𝑦𝑎 / 𝑥𝐶)
3634, 35sylan9eq 2819 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐶 = 𝑏 / 𝑦𝑎 / 𝑥𝐶)
3728, 29, 32, 33, 36cbvmpt2 6936 . . . . 5 (𝑥𝑋, 𝑦𝑌𝐶) = (𝑎𝑋, 𝑏𝑌𝑏 / 𝑦𝑎 / 𝑥𝐶)
3827, 37eqtri 2787 . . . 4 𝐹 = (𝑎𝑋, 𝑏𝑌𝑏 / 𝑦𝑎 / 𝑥𝐶)
3931nfel1 2922 . . . . . . 7 𝑥𝑎 / 𝑥𝐶𝑉
4033nfel1 2922 . . . . . . 7 𝑦𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉
4134eleq1d 2829 . . . . . . 7 (𝑥 = 𝑎 → (𝐶𝑉𝑎 / 𝑥𝐶𝑉))
4235eleq1d 2829 . . . . . . 7 (𝑦 = 𝑏 → (𝑎 / 𝑥𝐶𝑉𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉))
4339, 40, 41, 42rspc2 3473 . . . . . 6 ((𝑎𝑋𝑏𝑌) → (∀𝑥𝑋𝑦𝑌 𝐶𝑉𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉))
4411, 43mpan9 502 . . . . 5 ((𝜑 ∧ (𝑎𝑋𝑏𝑌)) → 𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉)
4544ralrimivva 3118 . . . 4 (𝜑 → ∀𝑎𝑋𝑏𝑌 𝑏 / 𝑦𝑎 / 𝑥𝐶𝑉)
461ne0d 4088 . . . 4 (𝜑𝑌 ≠ ∅)
47 fvmpt2curryd.y . . . 4 (𝜑𝑌𝑊)
4838, 45, 46, 47, 10mpt2curryvald 7603 . . 3 (𝜑 → (curry 𝐹𝐴) = (𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶))
4948fveq1d 6381 . 2 (𝜑 → ((curry 𝐹𝐴)‘𝐵) = ((𝑏𝑌𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)‘𝐵))
5027a1i 11 . . 3 (𝜑𝐹 = (𝑥𝑋, 𝑦𝑌𝐶))
51 csbco 3703 . . . . . . . 8 𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑦 / 𝑦𝑎 / 𝑥𝐶
52 csbid 3701 . . . . . . . 8 𝑦 / 𝑦𝑎 / 𝑥𝐶 = 𝑎 / 𝑥𝐶
5351, 52eqtr2i 2788 . . . . . . 7 𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶
5453a1i 11 . . . . . 6 (𝜑𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶)
5554csbeq2dv 4155 . . . . 5 (𝜑𝑥 / 𝑎𝑎 / 𝑥𝐶 = 𝑥 / 𝑎𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶)
56 csbco 3703 . . . . . 6 𝑥 / 𝑎𝑎 / 𝑥𝐶 = 𝑥 / 𝑥𝐶
57 csbid 3701 . . . . . 6 𝑥 / 𝑥𝐶 = 𝐶
5856, 57eqtri 2787 . . . . 5 𝑥 / 𝑎𝑎 / 𝑥𝐶 = 𝐶
59 csbcom 4157 . . . . 5 𝑥 / 𝑎𝑦 / 𝑏𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
6055, 58, 593eqtr3g 2822 . . . 4 (𝜑𝐶 = 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
61 csbeq1 3696 . . . . . . 7 (𝑥 = 𝐴𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6261adantr 472 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6362csbeq2dv 4155 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑦 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
64 csbeq1 3696 . . . . . 6 (𝑦 = 𝐵𝑦 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6564adantl 473 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6663, 65eqtrd 2799 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 / 𝑏𝑥 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
6760, 66sylan9eq 2819 . . 3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐶 = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
68 eqidd 2766 . . 3 ((𝜑𝑥 = 𝐴) → 𝑌 = 𝑌)
69 nfv 2009 . . 3 𝑥𝜑
70 nfv 2009 . . 3 𝑦𝜑
71 nfcv 2907 . . 3 𝑦𝐴
72 nfcv 2907 . . 3 𝑥𝐵
73 nfcv 2907 . . . . 5 𝑥𝐴
7473, 32nfcsb 3711 . . . 4 𝑥𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
7572, 74nfcsb 3711 . . 3 𝑥𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
769, 14nfcxfr 2905 . . 3 𝑦𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶
7750, 67, 68, 10, 1, 23, 69, 70, 71, 72, 75, 76ovmpt2dxf 6988 . 2 (𝜑 → (𝐴𝐹𝐵) = 𝐵 / 𝑏𝐴 / 𝑎𝑏 / 𝑦𝑎 / 𝑥𝐶)
7826, 49, 773eqtr4d 2809 1 (𝜑 → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  csb 3693  cmpt 4890  cfv 6070  (class class class)co 6846  cmpt2 6848  curry ccur 7598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-cur 7600
This theorem is referenced by:  pmatcollpw3lem  20881  logbfval  24833
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