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Theorem curry2f 7219
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 6758 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝑥𝐴𝐶𝐵) → (𝑥𝐹𝐶) ∈ 𝐷)
213com23 1268 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵𝑥𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
323expa 1262 . . 3 (((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) ∧ 𝑥𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
4 eqid 2626 . . 3 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶))
53, 4fmptd 6341 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → (𝑥𝐴 ↦ (𝑥𝐹𝐶)):𝐴𝐷)
6 ffn 6004 . . . 4 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
7 curry2.1 . . . . 5 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
87curry2 7218 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
96, 8sylan 488 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
109feq1d 5989 . 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 384   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153  cmpt 4678   × cxp 5077  ccnv 5078  cres 5081  ccom 5083   Fn wfn 5845  wf 5846  (class class class)co 6605  1st c1st 7114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-1st 7116  df-2nd 7117
This theorem is referenced by:  curry2ima  29320
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