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Mirrors > Home > ILE Home > Th. List > funres11 | 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 4838 | . 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
2 | cnvss 4707 | . 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
3 | funss 5137 | . 2 ⊢ (◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 → (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴))) | |
4 | 1, 2, 3 | mp2b 8 | 1 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3066 ◡ccnv 4533 ↾ cres 4536 Fun wfun 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-br 3925 df-opab 3985 df-rel 4541 df-cnv 4542 df-co 4543 df-res 4546 df-fun 5120 |
This theorem is referenced by: f1ssres 5332 resdif 5382 ssdomg 6665 sbthlemi8 6845 |
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