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Theorem curry2val 7259
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 7257 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
32fveq1d 6180 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷))
4 eqid 2620 . . . . . . . . . . 11 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶))
54dmmptss 5619 . . . . . . . . . 10 dom (𝑥𝐴 ↦ (𝑥𝐹𝐶)) ⊆ 𝐴
65sseli 3591 . . . . . . . . 9 (𝐷 ∈ dom (𝑥𝐴 ↦ (𝑥𝐹𝐶)) → 𝐷𝐴)
76con3i 150 . . . . . . . 8 𝐷𝐴 → ¬ 𝐷 ∈ dom (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
8 ndmfv 6205 . . . . . . . 8 𝐷 ∈ dom (𝑥𝐴 ↦ (𝑥𝐹𝐶)) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
97, 8syl 17 . . . . . . 7 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
109adantl 482 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
11 fndm 5978 . . . . . . 7 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
12 simpl 473 . . . . . . . 8 ((𝐷𝐴𝐶𝐵) → 𝐷𝐴)
1312con3i 150 . . . . . . 7 𝐷𝐴 → ¬ (𝐷𝐴𝐶𝐵))
14 ndmovg 6802 . . . . . . 7 ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷𝐴𝐶𝐵)) → (𝐷𝐹𝐶) = ∅)
1511, 13, 14syl2an 494 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → (𝐷𝐹𝐶) = ∅)
1610, 15eqtr4d 2657 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
1716ex 450 . . . 4 (𝐹 Fn (𝐴 × 𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
1817adantr 481 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
19 oveq1 6642 . . . 4 (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶))
20 ovex 6663 . . . 4 (𝐷𝐹𝐶) ∈ V
2119, 4, 20fvmpt 6269 . . 3 (𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
2218, 21pm2.61d2 172 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
233, 22eqtrd 2654 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = (𝐷𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  {csn 4168  cmpt 4720   × cxp 5102  ccnv 5103  dom cdm 5104  cres 5106  ccom 5108   Fn wfn 5871  cfv 5876  (class class class)co 6635  1st c1st 7151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-1st 7153  df-2nd 7154
This theorem is referenced by:  curry2ima  29460
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