Theorem fvmptndm 7028

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 5198	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	1, 2	eqtri 2789	. 2 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	3	fvopab4ndm 7027	1 ⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∅c0 4289  {copab 5178   ↦ cmpt 5197  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-dm 5676  df-iota 6499  df-fv 6551

This theorem is used by:  bropfvvvvlem  8095  bropfvvvv  8096  curry1val  8109  curry2val  8113  indval0  12240  homarcl  18110  arwval  18125  coafval  18146  pcofval  25206  newval  28065  leftval  28079  rightval  28080  fvmptrab  48070  fpprbasnn  48535  setrec2mpt  50516