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 4771 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | dmss 4797 | . . 3 ⊢ (◡𝐴 ⊆ ◡𝐵 → dom ◡𝐴 ⊆ dom ◡𝐵) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ⊆ 𝐵 → dom ◡𝐴 ⊆ dom ◡𝐵) |
4 | df-rn 4609 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | df-rn 4609 | . 2 ⊢ ran 𝐵 = dom ◡𝐵 | |
6 | 3, 4, 5 | 3sstr4g 3180 | 1 ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3111 ◡ccnv 4597 dom cdm 4598 ran crn 4599 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: imass1 4973 imass2 4974 rnxpss2 5031 ssxpbm 5033 ssxp2 5035 ssrnres 5040 funssxp 5351 fssres 5357 dff2 5623 fliftf 5761 1stcof 6123 2ndcof 6124 smores 6251 tfrcllembfn 6316 caserel 7043 frecuzrdgtcl 10337 lmss 12793 txss12 12813 txbasval 12814 |
Copyright terms: Public domain | W3C validator |