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Theorem curry2f 7433
 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 fovrn 6961 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝑥𝐴𝐶𝐵) → (𝑥𝐹𝐶) ∈ 𝐷)
213com23 1120 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵𝑥𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
323expa 1111 . . 3 (((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) ∧ 𝑥𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
4 eqid 2752 . . 3 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶))
53, 4fmptd 6540 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → (𝑥𝐴 ↦ (𝑥𝐹𝐶)):𝐴𝐷)
6 ffn 6198 . . . 4 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
7 curry2.1 . . . . 5 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
87curry2 7432 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
96, 8sylan 489 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
109feq1d 6183 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → (𝐺:𝐴𝐷 ↔ (𝑥𝐴 ↦ (𝑥𝐹𝐶)):𝐴𝐷))
115, 10mpbird 247 1 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺:𝐴𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1624   ∈ wcel 2131  Vcvv 3332  {csn 4313   ↦ cmpt 4873   × cxp 5256  ◡ccnv 5257   ↾ cres 5260   ∘ ccom 5262   Fn wfn 6036  ⟶wf 6037  (class class class)co 6805  1st c1st 7323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-1st 7325  df-2nd 7326 This theorem is referenced by:  curry2ima  29787
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