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Theorem mpt2curryvald 7601
Description: The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.)
Hypotheses
Ref Expression
mpt2curryd.f 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
mpt2curryd.c (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
mpt2curryd.n (𝜑𝑌 ≠ ∅)
mpt2curryvald.y (𝜑𝑌𝑊)
mpt2curryvald.a (𝜑𝐴𝑋)
Assertion
Ref Expression
mpt2curryvald (𝜑 → (curry 𝐹𝐴) = (𝑦𝑌𝐴 / 𝑥𝐶))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem mpt2curryvald
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 mpt2curryd.f . . . 4 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
2 mpt2curryd.c . . . 4 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
3 mpt2curryd.n . . . 4 (𝜑𝑌 ≠ ∅)
41, 2, 3mpt2curryd 7600 . . 3 (𝜑 → curry 𝐹 = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
5 nfcv 2907 . . . 4 𝑎(𝑦𝑌𝐶)
6 nfcv 2907 . . . . 5 𝑥𝑌
7 nfcsb1v 3709 . . . . 5 𝑥𝑎 / 𝑥𝐶
86, 7nfmpt 4907 . . . 4 𝑥(𝑦𝑌𝑎 / 𝑥𝐶)
9 csbeq1a 3702 . . . . 5 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
109mpteq2dv 4906 . . . 4 (𝑥 = 𝑎 → (𝑦𝑌𝐶) = (𝑦𝑌𝑎 / 𝑥𝐶))
115, 8, 10cbvmpt 4910 . . 3 (𝑥𝑋 ↦ (𝑦𝑌𝐶)) = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶))
124, 11syl6eq 2815 . 2 (𝜑 → curry 𝐹 = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶)))
13 csbeq1 3696 . . . 4 (𝑎 = 𝐴𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1413adantl 473 . . 3 ((𝜑𝑎 = 𝐴) → 𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1514mpteq2dv 4906 . 2 ((𝜑𝑎 = 𝐴) → (𝑦𝑌𝑎 / 𝑥𝐶) = (𝑦𝑌𝐴 / 𝑥𝐶))
16 mpt2curryvald.a . 2 (𝜑𝐴𝑋)
17 mpt2curryvald.y . . 3 (𝜑𝑌𝑊)
18 mptexg 6679 . . 3 (𝑌𝑊 → (𝑦𝑌𝐴 / 𝑥𝐶) ∈ V)
1917, 18syl 17 . 2 (𝜑 → (𝑦𝑌𝐴 / 𝑥𝐶) ∈ V)
2012, 15, 16, 19fvmptd 6479 1 (𝜑 → (curry 𝐹𝐴) = (𝑦𝑌𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  Vcvv 3350  csb 3693  c0 4081  cmpt 4890  cfv 6070  cmpt2 6846  curry ccur 7596
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 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  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 6847  df-oprab 6848  df-mpt2 6849  df-1st 7368  df-2nd 7369  df-cur 7598
This theorem is referenced by:  fvmpt2curryd  7602  pmatcollpw3lem  20870  logbmpt  24820
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