![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rnssi | Structured version Visualization version GIF version |
Description: Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.) |
Ref | Expression |
---|---|
rnssi.1 | ⊢ 𝐴 ⊆ 𝐵 |
Ref | Expression |
---|---|
rnssi | ⊢ ran 𝐴 ⊆ ran 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnssi.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | rnss 5953 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ran 𝐴 ⊆ ran 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3963 ran crn 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: rnresss 6037 ssrnres 6200 fssres 6775 smores 8391 rnttrcl 9760 brdom4 10568 smobeth 10624 nqerf 10968 catcoppccl 18171 catcoppcclOLD 18172 lern 18649 gsumzres 19942 gsumzaddlem 19954 gsumzadd 19955 dprdfadd 20055 txkgen 23676 dvlog 26708 perpln2 28734 pfxrn2 32909 fixssrn 35889 cnvrcl0 43615 |
Copyright terms: Public domain | W3C validator |