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Theorem fvmptndm 6848
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 5136 . . 3 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
31, 2eqtri 2765 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43fvopab4ndm 6847 1 𝑋𝐴 → (𝐹𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wcel 2110  c0 4237  {copab 5115  cmpt 5135  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-dm 5561  df-iota 6338  df-fv 6388
This theorem is referenced by:  bropfvvvvlem  7859  bropfvvvv  7860  curry1val  7873  curry2val  7877  homarcl  17534  arwval  17549  coafval  17570  pcofval  23907  newval  33776  leftval  33784  rightval  33785  fvmptrab  44456  fpprbasnn  44854
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