Theorem mpondm0 7383

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 7158	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
3	1, 2	eqtri 2843	. . . 4 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
4	3	dmeqi 5770	. . 3 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
5		dmoprabss 7253	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌)
6	4, 5	eqsstri 3998	. 2 ⊢ dom 𝐹 ⊆ (𝑋 × 𝑌)
7		nssdmovg 7327	. 2 ⊢ ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (𝑉𝐹𝑊) = ∅)
8	6, 7	mpan 688	1 ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ⊆ wss 3933  ∅c0 4288   × cxp 5550  dom cdm 5552  (class class class)co 7153  {coprab 7154   ∈ cmpo 7155

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3495  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4836  df-br 5064  df-opab 5126  df-xp 5558  df-dm 5562  df-iota 6311  df-fv 6360  df-ov 7156  df-oprab 7157  df-mpo 7158

This theorem is referenced by:  2mpo0  7391  elovmpt3imp  7399  el2mpocsbcl  7777  bropopvvv  7782  supp0prc  7830  brovex  7885  swrdnznd  14000  pfxnndmnd  14030  fullfunc  17172  fthfunc  17173  natfval  17212  evlval  20304  matbas0  21015  matrcl  21017  marrepfval  21165  marepvfval  21170  submafval  21184  minmar1fval  21251  hmeofval  22362  nghmfval  23327  wspthsn  27624  iswwlksnon  27629  iswspthsnon  27632  clwwlkn  27802  clwwlkneq0  27805  clwwlknon  27867  clwwlk0on0  27869  clwwlknon0  27870