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Theorem curry2val 7660
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 7658 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
32fveq1d 6540 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷))
4 eqid 2795 . . . . . . . 8 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶))
54fvmptndm 6663 . . . . . . 7 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
65adantl 482 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
7 fndm 6325 . . . . . . 7 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8 simpl 483 . . . . . . . 8 ((𝐷𝐴𝐶𝐵) → 𝐷𝐴)
98con3i 157 . . . . . . 7 𝐷𝐴 → ¬ (𝐷𝐴𝐶𝐵))
10 ndmovg 7187 . . . . . . 7 ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷𝐴𝐶𝐵)) → (𝐷𝐹𝐶) = ∅)
117, 9, 10syl2an 595 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → (𝐷𝐹𝐶) = ∅)
126, 11eqtr4d 2834 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷𝐴) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
1312ex 413 . . . 4 (𝐹 Fn (𝐴 × 𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
1413adantr 481 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (¬ 𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
15 oveq1 7023 . . . 4 (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶))
16 ovex 7048 . . . 4 (𝐷𝐹𝐶) ∈ V
1715, 4, 16fvmpt 6635 . . 3 (𝐷𝐴 → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
1814, 17pm2.61d2 182 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((𝑥𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
193, 18eqtrd 2831 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = (𝐷𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1522  wcel 2081  Vcvv 3437  c0 4211  {csn 4472  cmpt 5041   × cxp 5441  ccnv 5442  dom cdm 5443  cres 5445  ccom 5447   Fn wfn 6220  cfv 6225  (class class class)co 7016  1st c1st 7543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-1st 7545  df-2nd 7546
This theorem is referenced by:  curry2ima  30132
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