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Theorem curry2f 7814
 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 6503 . . 3 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
2 curry2.1 . . . 4 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
32curry2 7813 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
41, 3sylan 583 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
5 fovrn 7320 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝑥𝐴𝐶𝐵) → (𝑥𝐹𝐶) ∈ 𝐷)
653com23 1123 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵𝑥𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
763expa 1115 . 2 (((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) ∧ 𝑥𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
84, 7fmpt3d 6877 1 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺:𝐴𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4525   ↦ cmpt 5116   × cxp 5526  ◡ccnv 5527   ↾ cres 5530   ∘ ccom 5532   Fn wfn 6335  ⟶wf 6336  (class class class)co 7156  1st c1st 7697 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-1st 7699  df-2nd 7700 This theorem is referenced by:  curry2ima  30578
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