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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 7436 | . . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | eqtri 2763 | . . . 4 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} |
4 | 3 | dmeqi 5918 | . . 3 ⊢ dom 𝐹 = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} |
5 | dmoprabss 7536 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌) | |
6 | 4, 5 | eqsstri 4030 | . 2 ⊢ dom 𝐹 ⊆ (𝑋 × 𝑌) |
7 | nssdmovg 7615 | . 2 ⊢ ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (𝑉𝐹𝑊) = ∅) | |
8 | 6, 7 | mpan 690 | 1 ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∅c0 4339 × cxp 5687 dom cdm 5689 (class class class)co 7431 {coprab 7432 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: 2mpo0 7682 elovmpt3imp 7690 el2mpocsbcl 8109 bropopvvv 8114 supp0prc 8187 brovex 8246 swrdnznd 14677 pfxnndmnd 14707 fullfunc 17960 fthfunc 17961 natfval 18001 evlval 22137 matbas0 22430 matrcl 22432 marrepfval 22582 marepvfval 22587 submafval 22601 minmar1fval 22668 hmeofval 23782 nghmfval 24759 wspthsn 29878 iswwlksnon 29883 iswspthsnon 29886 clwwlkn 30055 clwwlkneq0 30058 clwwlknon 30119 clwwlk0on0 30121 clwwlknon0 30122 naryfval 48478 naryfvalixp 48479 |
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