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Theorem curry1f 7788
 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 6491 . . 3 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
2 curry1.1 . . . 4 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
32curry1 7786 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
41, 3sylan 583 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
5 fovrn 7302 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
653expa 1115 . 2 (((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) ∧ 𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
74, 6fmpt3d 6861 1 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺:𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444  {csn 4528   ↦ cmpt 5113   × cxp 5521  ◡ccnv 5522   ↾ cres 5525   ∘ ccom 5527   Fn wfn 6323  ⟶wf 6324  (class class class)co 7139  2nd c2nd 7674 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-1st 7675  df-2nd 7676 This theorem is referenced by:  nvinvfval  28427
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