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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 6011 | . 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
2 | cnvss 5879 | . 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
3 | funss 6577 | . 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 3949 ◡ccnv 5681 ↾ cres 5684 Fun wfun 6547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-in 3956 df-ss 3966 df-br 5153 df-opab 5215 df-rel 5689 df-cnv 5690 df-co 5691 df-res 5694 df-fun 6555 |
This theorem is referenced by: f1ssres 6806 resdif 6865 f1ssf1 6876 ssdomg 9027 sbthlem8 9121 spthispth 29560 |
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