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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 5342 | . 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
2 | cnvss 5216 | . 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
3 | funss 5822 | . 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 3540 ◡ccnv 5037 ↾ cres 5040 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-res 5050 df-fun 5806 |
This theorem is referenced by: f1ssres 6021 resdif 6070 ssdomg 7887 sbthlem8 7962 spthispth 26103 f1ssf1 40328 sPthisPth 40932 |
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