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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 7161 | . . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | eqtri 2844 | . . . 4 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} |
4 | 3 | dmeqi 5773 | . . 3 ⊢ dom 𝐹 = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} |
5 | dmoprabss 7256 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌) | |
6 | 4, 5 | eqsstri 4001 | . 2 ⊢ dom 𝐹 ⊆ (𝑋 × 𝑌) |
7 | nssdmovg 7330 | . 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 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 × cxp 5553 dom cdm 5555 (class class class)co 7156 {coprab 7157 ∈ cmpo 7158 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-dm 5565 df-iota 6314 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: 2mpo0 7394 elovmpt3imp 7402 el2mpocsbcl 7780 bropopvvv 7785 supp0prc 7833 brovex 7888 swrdnznd 14004 pfxnndmnd 14034 fullfunc 17176 fthfunc 17177 natfval 17216 evlval 20308 matbas0 21019 matrcl 21021 marrepfval 21169 marepvfval 21174 submafval 21188 minmar1fval 21255 hmeofval 22366 nghmfval 23331 wspthsn 27626 iswwlksnon 27631 iswspthsnon 27634 clwwlkn 27804 clwwlkneq0 27807 clwwlknon 27869 clwwlk0on0 27871 clwwlknon0 27872 |
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