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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 3086 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | 1 | eximdv 1852 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵)) |
3 | vex 2684 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 3 | eldm2 4732 | . . 3 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
5 | 3 | eldm2 4732 | . . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐵) |
6 | 2, 4, 5 | 3imtr4g 204 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ dom 𝐴 → 𝑥 ∈ dom 𝐵)) |
7 | 6 | ssrdv 3098 | 1 ⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1468 ∈ wcel 1480 ⊆ wss 3066 〈cop 3525 dom cdm 4534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-dm 4544 |
This theorem is referenced by: dmeq 4734 dmv 4750 rnss 4764 dmiin 4780 dmxpss2 4966 ssxpbm 4969 ssxp1 4970 cocnvres 5058 relrelss 5060 funssxp 5287 fvun1 5480 fndmdif 5518 fneqeql2 5522 tposss 6136 smores 6182 smores2 6184 tfrlemibfn 6218 tfrlemiubacc 6220 tfr1onlembfn 6234 tfr1onlemubacc 6236 tfr1onlemres 6239 tfrcllembfn 6247 tfrcllemubacc 6249 tfrcllemres 6252 frecuzrdgtcl 10178 frecuzrdgdomlem 10183 ennnfonelemex 11916 strleund 12036 strleun 12037 dvbssntrcntop 12811 |
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