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Theorem curry2f 8132
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 6737 . . 3 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
2 curry2.1 . . . 4 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
32curry2 8131 . . 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 1125 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵𝑥𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
763expa 1117 . 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 1537  wcel 2106  Vcvv 3478  {csn 4631  cmpt 5231   × cxp 5687  ccnv 5688  cres 5691  ccom 5693   Fn wfn 6558  wf 6559  (class class class)co 7431  1st c1st 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-1st 8013  df-2nd 8014
This theorem is referenced by:  curry2ima  32724
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