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Theorem curry2val 8133
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 8131 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
32fveq1d 6909 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷))
4 eqid 2735 . . . . . . . 8 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶))
54fvmptndm 7047 . . . . . . 7 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
65adantl 481 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
7 fndm 6672 . . . . . . 7 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8 simpl 482 . . . . . . . 8 ((𝐷𝐴𝐶𝐵) → 𝐷𝐴)
98con3i 154 . . . . . . 7 𝐷𝐴 → ¬ (𝐷𝐴𝐶𝐵))
10 ndmovg 7616 . . . . . . 7 ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷𝐴𝐶𝐵)) → (𝐷𝐹𝐶) = ∅)
117, 9, 10syl2an 596 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → (𝐷𝐹𝐶) = ∅)
126, 11eqtr4d 2778 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
1312ex 412 . . . 4 (𝐹 Fn (𝐴 × 𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
1413adantr 480 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
15 oveq1 7438 . . . 4 (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶))
16 ovex 7464 . . . 4 (𝐷𝐹𝐶) ∈ V
1715, 4, 16fvmpt 7016 . . 3 (𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
1814, 17pm2.61d2 181 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
193, 18eqtrd 2775 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = (𝐷𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {csn 4631  cmpt 5231   × cxp 5687  ccnv 5688  dom cdm 5689  cres 5691  ccom 5693   Fn wfn 6558  cfv 6563  (class class class)co 7431  1st c1st 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-1st 8013  df-2nd 8014
This theorem is referenced by:  curry2ima  32724
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