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Theorem curry2f 8133
Description: Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
Hypothesis
Ref Expression
curry2.1 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
Assertion
Ref Expression
curry2f ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺:𝐴𝐷)

Proof of Theorem curry2f
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ffn 6736 . . 3 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
2 curry2.1 . . . 4 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
32curry2 8132 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
41, 3sylan 580 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
5 fovcdm 7603 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝑥𝐴𝐶𝐵) → (𝑥𝐹𝐶) ∈ 𝐷)
653com23 1127 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵𝑥𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
763expa 1119 . 2 (((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) ∧ 𝑥𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
84, 7fmpt3d 7136 1 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺:𝐴𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cmpt 5225   × cxp 5683  ccnv 5684  cres 5687  ccom 5689   Fn wfn 6556  wf 6557  (class class class)co 7431  1st c1st 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015
This theorem is referenced by:  curry2ima  32718
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