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Mirrors > Home > MPE Home > Th. List > mpt2ndm0 | 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 |
---|---|
mpt2ndm0.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
Ref | Expression |
---|---|
mpt2ndm0 | ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2ndm0.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) | |
2 | df-mpt2 6655 | . . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | eqtri 2644 | . . . 4 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} |
4 | 3 | dmeqi 5325 | . . 3 ⊢ dom 𝐹 = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} |
5 | dmoprabss 6742 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌) | |
6 | 4, 5 | eqsstri 3635 | . 2 ⊢ dom 𝐹 ⊆ (𝑋 × 𝑌) |
7 | nssdmovg 6816 | . 2 ⊢ ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (𝑉𝐹𝑊) = ∅) | |
8 | 6, 7 | mpan 706 | 1 ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∅c0 3915 × cxp 5112 dom cdm 5114 (class class class)co 6650 {coprab 6651 ↦ cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: 2mpt20 6882 elovmpt3imp 6890 el2mpt2csbcl 7250 bropopvvv 7255 supp0prc 7298 brovex 7348 fullfunc 16566 fthfunc 16567 natfval 16606 evlval 19524 matbas0 20216 matrcl 20218 marrepfval 20366 marepvfval 20371 submafval 20385 minmar1fval 20452 hmeofval 21561 nghmfval 22526 wspthsn 26735 iswwlksnon 26740 iswspthsnon 26741 wspn0 26820 numclwwlkffin0 27215 |
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