Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > rnss | GIF version |
Description: Subset theorem for range. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
rnss | ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss 4682 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | dmss 4708 | . . 3 ⊢ (◡𝐴 ⊆ ◡𝐵 → dom ◡𝐴 ⊆ dom ◡𝐵) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ⊆ 𝐵 → dom ◡𝐴 ⊆ dom ◡𝐵) |
4 | df-rn 4520 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | df-rn 4520 | . 2 ⊢ ran 𝐵 = dom ◡𝐵 | |
6 | 3, 4, 5 | 3sstr4g 3110 | 1 ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3041 ◡ccnv 4508 dom cdm 4509 ran crn 4510 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-cnv 4517 df-dm 4519 df-rn 4520 |
This theorem is referenced by: imass1 4884 imass2 4885 rnxpss2 4942 ssxpbm 4944 ssxp2 4946 ssrnres 4951 funssxp 5262 fssres 5268 dff2 5532 fliftf 5668 1stcof 6029 2ndcof 6030 smores 6157 tfrcllembfn 6222 caserel 6940 frecuzrdgtcl 10153 lmss 12342 txss12 12362 txbasval 12363 |
Copyright terms: Public domain | W3C validator |