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Theorem fvmptndm 6975
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 5187 . . 3 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
31, 2eqtri 2765 . 2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43fvopab4ndm 6974 1 𝑋𝐴 → (𝐹𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  c0 4280  {copab 5165  cmpt 5186  cfv 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-dm 5641  df-iota 6445  df-fv 6501
This theorem is referenced by:  bropfvvvvlem  8015  bropfvvvv  8016  curry1val  8029  curry2val  8033  homarcl  17874  arwval  17889  coafval  17910  pcofval  24325  newval  27137  leftval  27145  rightval  27146  fvmptrab  45425  fpprbasnn  45822  setrec2mpt  47043
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