Theorem fvmptndm 7002

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 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fvmptndm	⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅)

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.

Step	Hyp	Ref	Expression
1		fvmptndm.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2		df-mpt 5192	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	1, 2	eqtri 2753	. 2 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	3	fvopab4ndm 7001	1 ⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∅c0 4299  {copab 5172   ↦ cmpt 5191  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-dm 5651  df-iota 6467  df-fv 6522

This theorem is referenced by:  bropfvvvvlem  8073  bropfvvvv  8074  curry1val  8087  curry2val  8091  homarcl  17997  arwval  18012  coafval  18033  pcofval  24917  newval  27770  leftval  27778  rightval  27779  fvmptrab  47297  fpprbasnn  47734  setrec2mpt  49690