Theorem mpondm0 7586

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 7351	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
3	1, 2	eqtri 2754	. . . 4 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
4	3	dmeqi 5844	. . 3 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
5		dmoprabss 7450	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌)
6	4, 5	eqsstri 3981	. 2 ⊢ dom 𝐹 ⊆ (𝑋 × 𝑌)
7		nssdmovg 7528	. 2 ⊢ ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (𝑉𝐹𝑊) = ∅)
8	6, 7	mpan 690	1 ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3902  ∅c0 4283   × cxp 5614  dom cdm 5616  (class class class)co 7346  {coprab 7347   ∈ cmpo 7348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-xp 5622  df-dm 5626  df-iota 6437  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351

This theorem is referenced by:  2mpo0  7595  elovmpt3imp  7603  el2mpocsbcl  8015  bropopvvv  8020  supp0prc  8093  brovex  8152  swrdnznd  14547  pfxnndmnd  14577  fullfunc  17812  fthfunc  17813  natfval  17853  evlval  22028  matbas0  22323  matrcl  22325  marrepfval  22473  marepvfval  22478  submafval  22492  minmar1fval  22559  hmeofval  23671  nghmfval  24635  wspthsn  29824  iswwlksnon  29829  iswspthsnon  29832  clwwlkn  30001  clwwlkneq0  30004  clwwlknon  30065  clwwlk0on0  30067  clwwlknon0  30068  fineqvnttrclselem1  35129  naryfval  48659  naryfvalixp  48660  oppc1stflem  49318