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Theorem fvmptndm 6347
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 4763 . . 3 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
31, 2eqtri 2673 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43fvopab4ndm 6346 1 𝑋𝐴 → (𝐹𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  c0 3948  {copab 4745  cmpt 4762  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-dm 5153  df-iota 5889  df-fv 5934
This theorem is referenced by:  bropfvvvvlem  7301  bropfvvvv  7302  homarcl  16725  fvmptrab  41631
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