Theorem mpt2ndm0 7135

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
mpt2ndm0.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)

Assertion

Ref	Expression
mpt2ndm0	⊢ (¬ (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 mpt2ndm0

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpt2ndm0.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
2		df-mpt2 6910	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
3	1, 2	eqtri 2849	. . . 4 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
4	3	dmeqi 5557	. . 3 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)}
5		dmoprabss 7002	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌)
6	4, 5	eqsstri 3860	. 2 ⊢ dom 𝐹 ⊆ (𝑋 × 𝑌)
7		nssdmovg 7076	. 2 ⊢ ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (𝑉𝐹𝑊) = ∅)
8	6, 7	mpan 683	1 ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ⊆ wss 3798  ∅c0 4144   × cxp 5340  dom cdm 5342  (class class class)co 6905  {coprab 6906   ↦ cmpt2 6907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-xp 5348  df-dm 5352  df-iota 6086  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910

This theorem is referenced by:  2mpt20  7142  elovmpt3imp  7150  el2mpt2csbcl  7513  bropopvvv  7519  supp0prc  7562  brovex  7613  swrdnznd  13702  pfxnndmnd  13751  fullfunc  16918  fthfunc  16919  natfval  16958  evlval  19884  matbas0  20583  matrcl  20585  marrepfval  20734  marepvfval  20739  submafval  20753  minmar1fval  20820  hmeofval  21932  nghmfval  22896  wspthsn  27147  iswwlksnon  27152  iswspthsnon  27155  clwwlkn  27366  clwwlkneq0  27371  clwwlknon  27461  clwwlk0on0  27465  clwwlknon0  27466