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Theorem fvmptndm 6533
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 4923 . . 3 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
31, 2eqtri 2821 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43fvopab4ndm 6532 1 𝑋𝐴 → (𝐹𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  c0 4115  {copab 4905  cmpt 4922  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-dm 5322  df-iota 6064  df-fv 6109
This theorem is referenced by:  bropfvvvvlem  7493  bropfvvvv  7494  homarcl  16992  fvmptrab  42147
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