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Theorem curry1f 7216
 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 fovrn 6757 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
213expa 1262 . . 3 (((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) ∧ 𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
3 eqid 2621 . . 3 (𝑥𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥𝐵 ↦ (𝐶𝐹𝑥))
42, 3fmptd 6340 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → (𝑥𝐵 ↦ (𝐶𝐹𝑥)):𝐵𝐷)
5 ffn 6002 . . . 4 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
6 curry1.1 . . . . 5 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
76curry1 7214 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
85, 7sylan 488 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
98feq1d 5987 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → (𝐺:𝐵𝐷 ↔ (𝑥𝐵 ↦ (𝐶𝐹𝑥)):𝐵𝐷))
104, 9mpbird 247 1 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺:𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  {csn 4148   ↦ cmpt 4673   × cxp 5072  ◡ccnv 5073   ↾ cres 5076   ∘ ccom 5078   Fn wfn 5842  ⟶wf 5843  (class class class)co 6604  2nd c2nd 7112 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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-1st 7113  df-2nd 7114 This theorem is referenced by:  nvinvfval  27341
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