![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > dmss | GIF version |
Description: Subset theorem for domain. (Contributed by NM, 11-Aug-1994.) |
Ref | Expression |
---|---|
dmss | ⊢ (A ⊆ B → dom A ⊆ dom B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2933 | . . . 4 ⊢ (A ⊆ B → (〈x, y〉 ∈ A → 〈x, y〉 ∈ B)) | |
2 | 1 | eximdv 1757 | . . 3 ⊢ (A ⊆ B → (∃y〈x, y〉 ∈ A → ∃y〈x, y〉 ∈ B)) |
3 | vex 2554 | . . . 4 ⊢ x ∈ V | |
4 | 3 | eldm2 4476 | . . 3 ⊢ (x ∈ dom A ↔ ∃y〈x, y〉 ∈ A) |
5 | 3 | eldm2 4476 | . . 3 ⊢ (x ∈ dom B ↔ ∃y〈x, y〉 ∈ B) |
6 | 2, 4, 5 | 3imtr4g 194 | . 2 ⊢ (A ⊆ B → (x ∈ dom A → x ∈ dom B)) |
7 | 6 | ssrdv 2945 | 1 ⊢ (A ⊆ B → dom A ⊆ dom B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1378 ∈ wcel 1390 ⊆ wss 2911 〈cop 3370 dom cdm 4288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-dm 4298 |
This theorem is referenced by: dmeq 4478 dmv 4494 rnss 4507 dmiin 4523 ssxpbm 4699 ssxp1 4700 relrelss 4787 funssxp 5003 fvun1 5182 fndmdif 5215 fneqeql2 5219 tposss 5802 smores 5848 smores2 5850 tfrlemibfn 5883 tfrlemiubacc 5885 frecuzrdgfn 8879 |
Copyright terms: Public domain | W3C validator |