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Theorem mpocurryvald 8272
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
mpocurryd.f 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
mpocurryd.c (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
mpocurryd.n (𝜑𝑌 ≠ ∅)
mpocurryvald.y (𝜑𝑌𝑊)
mpocurryvald.a (𝜑𝐴𝑋)
Assertion
Ref Expression
mpocurryvald (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 mpocurryvald
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 mpocurryd.f . . . 4 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
2 mpocurryd.c . . . 4 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
3 mpocurryd.n . . . 4 (𝜑𝑌 ≠ ∅)
41, 2, 3mpocurryd 8271 . . 3 (𝜑 → curry 𝐹 = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
5 nfcv 2892 . . . 4 𝑎(𝑦𝑌𝐶)
6 nfcv 2892 . . . . 5 𝑥𝑌
7 nfcsb1v 3910 . . . . 5 𝑥𝑎 / 𝑥𝐶
86, 7nfmpt 5250 . . . 4 𝑥(𝑦𝑌𝑎 / 𝑥𝐶)
9 csbeq1a 3899 . . . . 5 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
109mpteq2dv 5245 . . . 4 (𝑥 = 𝑎 → (𝑦𝑌𝐶) = (𝑦𝑌𝑎 / 𝑥𝐶))
115, 8, 10cbvmpt 5254 . . 3 (𝑥𝑋 ↦ (𝑦𝑌𝐶)) = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶))
124, 11eqtrdi 2781 . 2 (𝜑 → curry 𝐹 = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶)))
13 csbeq1 3888 . . . 4 (𝑎 = 𝐴𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1413adantl 480 . . 3 ((𝜑𝑎 = 𝐴) → 𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1514mpteq2dv 5245 . 2 ((𝜑𝑎 = 𝐴) → (𝑦𝑌𝑎 / 𝑥𝐶) = (𝑦𝑌𝐴 / 𝑥𝐶))
16 mpocurryvald.a . 2 (𝜑𝐴𝑋)
17 mpocurryvald.y . . 3 (𝜑𝑌𝑊)
1817mptexd 7231 . 2 (𝜑 → (𝑦𝑌𝐴 / 𝑥𝐶) ∈ V)
1912, 15, 16, 18fvmptd 7006 1 (𝜑 → (curry 𝐹𝐴) = (𝑦𝑌𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2930  wral 3051  Vcvv 3463  csb 3885  c0 4318  cmpt 5226  cfv 6542  cmpo 7417  curry ccur 8267
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-cur 8269
This theorem is referenced by:  fvmpocurryd  8273  pmatcollpw3lem  22701  logbmpt  26736
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