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Theorem fvmptndm 7060
Description: Value of a function given by the maps-to notation, outside of its domain. (Contributed by AV, 31-Dec-2020.)
Hypothesis
Ref Expression
fvmptndm.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmptndm 𝑋𝐴 → (𝐹𝑋) = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem fvmptndm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvmptndm.1 . . 3 𝐹 = (𝑥𝐴𝐵)
2 df-mpt 5250 . . 3 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
31, 2eqtri 2768 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43fvopab4ndm 7059 1 𝑋𝐴 → (𝐹𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2108  c0 4352  {copab 5228  cmpt 5249  cfv 6573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-dm 5710  df-iota 6525  df-fv 6581
This theorem is referenced by:  bropfvvvvlem  8132  bropfvvvv  8133  curry1val  8146  curry2val  8150  homarcl  18095  arwval  18110  coafval  18131  pcofval  25062  newval  27912  leftval  27920  rightval  27921  fvmptrab  47207  fpprbasnn  47603  setrec2mpt  48789
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