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Theorem curry1f 7477
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 7006 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
213expa 1147 . . 3 (((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) ∧ 𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
32fmpttd 6579 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → (𝑥𝐵 ↦ (𝐶𝐹𝑥)):𝐵𝐷)
4 ffn 6225 . . . 4 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
5 curry1.1 . . . . 5 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
65curry1 7475 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
74, 6sylan 575 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
87feq1d 6210 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → (𝐺:𝐵𝐷 ↔ (𝑥𝐵 ↦ (𝐶𝐹𝑥)):𝐵𝐷))
93, 8mpbird 248 1 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺:𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  {csn 4336  cmpt 4890   × cxp 5277  ccnv 5278  cres 5281  ccom 5283   Fn wfn 6065  wf 6066  (class class class)co 6846  2nd c2nd 7369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-1st 7370  df-2nd 7371
This theorem is referenced by:  nvinvfval  27972
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