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Theorem curry2val 7319
Description: The value 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
curry2val ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = (𝐷𝐹𝐶))

Proof of Theorem curry2val
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 curry2.1 . . . 4 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
21curry2 7317 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
32fveq1d 6231 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷))
4 eqid 2651 . . . . . . . . . . 11 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶))
54dmmptss 5669 . . . . . . . . . 10 dom (𝑥𝐴 ↦ (𝑥𝐹𝐶)) ⊆ 𝐴
65sseli 3632 . . . . . . . . 9 (𝐷 ∈ dom (𝑥𝐴 ↦ (𝑥𝐹𝐶)) → 𝐷𝐴)
76con3i 150 . . . . . . . 8 𝐷𝐴 → ¬ 𝐷 ∈ dom (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
8 ndmfv 6256 . . . . . . . 8 𝐷 ∈ dom (𝑥𝐴 ↦ (𝑥𝐹𝐶)) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
97, 8syl 17 . . . . . . 7 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
109adantl 481 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
11 fndm 6028 . . . . . . 7 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
12 simpl 472 . . . . . . . 8 ((𝐷𝐴𝐶𝐵) → 𝐷𝐴)
1312con3i 150 . . . . . . 7 𝐷𝐴 → ¬ (𝐷𝐴𝐶𝐵))
14 ndmovg 6859 . . . . . . 7 ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷𝐴𝐶𝐵)) → (𝐷𝐹𝐶) = ∅)
1511, 13, 14syl2an 493 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → (𝐷𝐹𝐶) = ∅)
1610, 15eqtr4d 2688 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
1716ex 449 . . . 4 (𝐹 Fn (𝐴 × 𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
1817adantr 480 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
19 oveq1 6697 . . . 4 (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶))
20 ovex 6718 . . . 4 (𝐷𝐹𝐶) ∈ V
2119, 4, 20fvmpt 6321 . . 3 (𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
2218, 21pm2.61d2 172 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
233, 22eqtrd 2685 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = (𝐷𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  {csn 4210  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  cres 5145  ccom 5147   Fn wfn 5921  cfv 5926  (class class class)co 6690  1st c1st 7208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-1st 7210  df-2nd 7211
This theorem is referenced by:  curry2ima  29614
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