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| Mirrors > Home > MPE Home > Th. List > funres11 | Structured version Visualization version GIF version | ||
| Description: The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.) | 
| Ref | Expression | 
|---|---|
| funres11 | ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | resss 6018 | . 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
| 2 | cnvss 5882 | . 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
| 3 | funss 6584 | . 2 ⊢ (◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 → (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴))) | |
| 4 | 1, 2, 3 | mp2b 10 | 1 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ⊆ wss 3950 ◡ccnv 5683 ↾ cres 5686 Fun wfun 6554 | 
| 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-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-in 3957 df-ss 3967 df-br 5143 df-opab 5205 df-rel 5691 df-cnv 5692 df-co 5693 df-res 5696 df-fun 6562 | 
| This theorem is referenced by: f1ssres 6810 resdif 6868 f1ssf1 6879 resf1extb 7957 ssdomg 9041 sbthlem8 9131 spthispth 29745 | 
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