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Theorem curry1f 7441
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 6971 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
213expa 1112 . . 3 (((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) ∧ 𝑥𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
3 eqid 2761 . . 3 (𝑥𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥𝐵 ↦ (𝐶𝐹𝑥))
42, 3fmptd 6550 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → (𝑥𝐵 ↦ (𝐶𝐹𝑥)):𝐵𝐷)
5 ffn 6207 . . . 4 (𝐹:(𝐴 × 𝐵)⟶𝐷𝐹 Fn (𝐴 × 𝐵))
6 curry1.1 . . . . 5 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
76curry1 7439 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
85, 7sylan 489 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
98feq1d 6192 . 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 383   = wceq 1632  wcel 2140  Vcvv 3341  {csn 4322  cmpt 4882   × cxp 5265  ccnv 5266  cres 5269  ccom 5271   Fn wfn 6045  wf 6046  (class class class)co 6815  2nd c2nd 7334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-1st 7335  df-2nd 7336
This theorem is referenced by:  nvinvfval  27826
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