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| 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 5949 | . 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 3950 ran crn 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 | 
| This theorem is referenced by: rnresss 6034 ssrnres 6197 fssres 6773 smores 8393 rnttrcl 9763 brdom4 10571 smobeth 10627 nqerf 10971 catcoppccl 18163 lern 18637 gsumzres 19928 gsumzaddlem 19940 gsumzadd 19941 dprdfadd 20041 txkgen 23661 dvlog 26694 perpln2 28720 pfxrn2 32925 fixssrn 35909 cnvrcl0 43643 | 
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