![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mpondm0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
mpondm0.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
Ref | Expression |
---|---|
mpondm0 | ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpondm0.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) | |
2 | df-mpo 7416 | . . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | eqtri 2760 | . . . 4 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} |
4 | 3 | dmeqi 5904 | . . 3 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} |
5 | dmoprabss 7513 | . . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌) | |
6 | 4, 5 | eqsstri 4016 | . 2 ⊢ dom 𝐹 ⊆ (𝑋 × 𝑌) |
7 | nssdmovg 7591 | . 2 ⊢ ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (𝑉𝐹𝑊) = ∅) | |
8 | 6, 7 | mpan 688 | 1 ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3948 ∅c0 4322 × cxp 5674 dom cdm 5676 (class class class)co 7411 {coprab 7412 ∈ cmpo 7413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-dm 5686 df-iota 6495 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 |
This theorem is referenced by: 2mpo0 7657 elovmpt3imp 7665 el2mpocsbcl 8073 bropopvvv 8078 supp0prc 8151 brovex 8209 swrdnznd 14596 pfxnndmnd 14626 fullfunc 17861 fthfunc 17862 natfval 17901 evlval 21877 matbas0 22130 matrcl 22132 marrepfval 22282 marepvfval 22287 submafval 22301 minmar1fval 22368 hmeofval 23482 nghmfval 24459 wspthsn 29357 iswwlksnon 29362 iswspthsnon 29365 clwwlkn 29534 clwwlkneq0 29537 clwwlknon 29598 clwwlk0on0 29600 clwwlknon0 29601 naryfval 47402 naryfvalixp 47403 |
Copyright terms: Public domain | W3C validator |