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Theorem mpt2curryvald 7283
 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 7282 . . 3 (𝜑 → curry 𝐹 = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
5 nfcv 2751 . . . 4 𝑎(𝑦𝑌𝐶)
6 nfcv 2751 . . . . 5 𝑥𝑌
7 nfcsb1v 3515 . . . . 5 𝑥𝑎 / 𝑥𝐶
86, 7nfmpt 4674 . . . 4 𝑥(𝑦𝑌𝑎 / 𝑥𝐶)
9 csbeq1a 3508 . . . . 5 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
109mpteq2dv 4673 . . . 4 (𝑥 = 𝑎 → (𝑦𝑌𝐶) = (𝑦𝑌𝑎 / 𝑥𝐶))
115, 8, 10cbvmpt 4677 . . 3 (𝑥𝑋 ↦ (𝑦𝑌𝐶)) = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶))
124, 11syl6eq 2660 . 2 (𝜑 → curry 𝐹 = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶)))
13 csbeq1 3502 . . . 4 (𝑎 = 𝐴𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1413adantl 481 . . 3 ((𝜑𝑎 = 𝐴) → 𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1514mpteq2dv 4673 . 2 ((𝜑𝑎 = 𝐴) → (𝑦𝑌𝑎 / 𝑥𝐶) = (𝑦𝑌𝐴 / 𝑥𝐶))
16 mpt2curryvald.a . 2 (𝜑𝐴𝑋)
17 mpt2curryvald.y . . 3 (𝜑𝑌𝑊)
18 mptexg 6389 . . 3 (𝑌𝑊 → (𝑦𝑌𝐴 / 𝑥𝐶) ∈ V)
1917, 18syl 17 . 2 (𝜑 → (𝑦𝑌𝐴 / 𝑥𝐶) ∈ V)
2012, 15, 16, 19fvmptd 6197 1 (𝜑 → (curry 𝐹𝐴) = (𝑦𝑌𝐴 / 𝑥𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ⦋csb 3499  ∅c0 3874   ↦ cmpt 4643  ‘cfv 5804   ↦ cmpt2 6551  curry ccur 7278 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cur 7280 This theorem is referenced by:  fvmpt2curryd  7284  pmatcollpw3lem  20407  logbmpt  24326
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