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Theorem curry2val 7791
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 7789 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
32fveq1d 6651 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷))
4 eqid 2801 . . . . . . . 8 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶))
54fvmptndm 6779 . . . . . . 7 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
65adantl 485 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
7 fndm 6429 . . . . . . 7 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8 simpl 486 . . . . . . . 8 ((𝐷𝐴𝐶𝐵) → 𝐷𝐴)
98con3i 157 . . . . . . 7 𝐷𝐴 → ¬ (𝐷𝐴𝐶𝐵))
10 ndmovg 7315 . . . . . . 7 ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷𝐴𝐶𝐵)) → (𝐷𝐹𝐶) = ∅)
117, 9, 10syl2an 598 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → (𝐷𝐹𝐶) = ∅)
126, 11eqtr4d 2839 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
1312ex 416 . . . 4 (𝐹 Fn (𝐴 × 𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
1413adantr 484 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
15 oveq1 7146 . . . 4 (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶))
16 ovex 7172 . . . 4 (𝐷𝐹𝐶) ∈ V
1715, 4, 16fvmpt 6749 . . 3 (𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
1814, 17pm2.61d2 184 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
193, 18eqtrd 2836 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = (𝐷𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  c0 4246  {csn 4528  cmpt 5113   × cxp 5521  ccnv 5522  dom cdm 5523  cres 5525  ccom 5527   Fn wfn 6323  cfv 6328  (class class class)co 7139  1st c1st 7673
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-1st 7675  df-2nd 7676
This theorem is referenced by:  curry2ima  30471
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