MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  curry1f Structured version   Visualization version   GIF version

Theorem curry1f 7648
Description: Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)
Hypothesis
Ref Expression
curry1.1 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
Assertion
Ref Expression
curry1f ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺:𝐵𝐷)

Proof of Theorem curry1f
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ffn 6374 . . 3 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
2 curry1.1 . . . 4 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
32curry1 7646 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
41, 3sylan 580 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
5 fovrn 7165 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
653expa 1109 . 2 (((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) ∧ 𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
74, 6fmpt3d 6734 1 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺:𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1520  wcel 2079  Vcvv 3432  {csn 4466  cmpt 5035   × cxp 5433  ccnv 5434  cres 5437  ccom 5439   Fn wfn 6212  wf 6213  (class class class)co 7007  2nd c2nd 7535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-1st 7536  df-2nd 7537
This theorem is referenced by:  nvinvfval  28096
  Copyright terms: Public domain W3C validator