Theorem mpondm0 7658

Description: The value of an operation given by a maps-to rule is the empty set if the arguments are not contained in the base sets of the rule. (Contributed by Alexander van der Vekens, 12-Oct-2017.)

Hypothesis

Ref	Expression
mpondm0.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)

Assertion

Ref	Expression
mpondm0	⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅)

Distinct variable groups:   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝑉(𝑥, 𝑦)   𝑊(𝑥, 𝑦)

Proof of Theorem mpondm0

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpondm0.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
2		df-mpo 7422	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
3	1, 2	eqtri 2785	. . . 4 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
4	3	dmeqi 5892	. . 3 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
5		dmoprabss 7521	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌)
6	4, 5	eqsstri 3980	. 2 ⊢ dom 𝐹 ⊆ (𝑋 × 𝑌)
7		nssdmovg 7600	. 2 ⊢ ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (𝑉𝐹𝑊) = ∅)
8	6, 7	mpan 703	1 ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3902  ∅c0 4282   × cxp 5657  dom cdm 5659  (class class class)co 7417  {coprab 7418   ∈ cmpo 7419

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-dm 5669  df-iota 6493  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422

This theorem is used by:  2mpo0  7667  elovmpt3imp  7675  el2mpocsbcl  8086  bropopvvv  8091  supp0prc  8165  brovex  8224  swrdnznd  14714  pfxnndmnd  14746  fullfunc  18003  fthfunc  18004  natfval  18044  evlval  22322  matbas0  22638  matrcl  22640  marrepfval  22788  marepvfval  22793  submafval  22807  minmar1fval  22874  hmeofval  23990  nghmfval  24954  wspthsn  30324  iswwlksnon  30329  iswspthsnon  30332  clwwlkn  30504  clwwlkneq0  30507  clwwlknon  30568  clwwlk0on0  30570  clwwlknon0  30571  fineqvnttrclselem1  35655  naryfval  49566  naryfvalixp  49567  oppc1stflem  50221