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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 5963 | . 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
2 | cnvss 5829 | . 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
3 | funss 6521 | . 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 3911 ◡ccnv 5633 ↾ cres 5636 Fun wfun 6491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-ss 3928 df-br 5107 df-opab 5169 df-rel 5641 df-cnv 5642 df-co 5643 df-res 5646 df-fun 6499 |
This theorem is referenced by: f1ssres 6747 resdif 6806 f1ssf1 6817 ssdomg 8943 sbthlem8 9037 spthispth 28716 |
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