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Theorem mpocurryvald 8205
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 8204 . . 3 (𝜑 → curry 𝐹 = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
5 nfcv 2904 . . . 4 𝑎(𝑦𝑌𝐶)
6 nfcv 2904 . . . . 5 𝑥𝑌
7 nfcsb1v 3884 . . . . 5 𝑥𝑎 / 𝑥𝐶
86, 7nfmpt 5216 . . . 4 𝑥(𝑦𝑌𝑎 / 𝑥𝐶)
9 csbeq1a 3873 . . . . 5 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
109mpteq2dv 5211 . . . 4 (𝑥 = 𝑎 → (𝑦𝑌𝐶) = (𝑦𝑌𝑎 / 𝑥𝐶))
115, 8, 10cbvmpt 5220 . . 3 (𝑥𝑋 ↦ (𝑦𝑌𝐶)) = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶))
124, 11eqtrdi 2789 . 2 (𝜑 → curry 𝐹 = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶)))
13 csbeq1 3862 . . . 4 (𝑎 = 𝐴𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1413adantl 483 . . 3 ((𝜑𝑎 = 𝐴) → 𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1514mpteq2dv 5211 . 2 ((𝜑𝑎 = 𝐴) → (𝑦𝑌𝑎 / 𝑥𝐶) = (𝑦𝑌𝐴 / 𝑥𝐶))
16 mpocurryvald.a . 2 (𝜑𝐴𝑋)
17 mpocurryvald.y . . 3 (𝜑𝑌𝑊)
1817mptexd 7178 . 2 (𝜑 → (𝑦𝑌𝐴 / 𝑥𝐶) ∈ V)
1912, 15, 16, 18fvmptd 6959 1 (𝜑 → (curry 𝐹𝐴) = (𝑦𝑌𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  Vcvv 3447  csb 3859  c0 4286  cmpt 5192  cfv 6500  cmpo 7363  curry ccur 8200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-cur 8202
This theorem is referenced by:  fvmpocurryd  8206  pmatcollpw3lem  22155  logbmpt  26161
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