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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 4813 | . 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
2 | cnvss 4682 | . 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
3 | funss 5112 | . 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 3041 ◡ccnv 4508 ↾ cres 4511 Fun wfun 5087 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 df-br 3900 df-opab 3960 df-rel 4516 df-cnv 4517 df-co 4518 df-res 4521 df-fun 5095 |
This theorem is referenced by: f1ssres 5307 resdif 5357 ssdomg 6640 sbthlemi8 6820 |
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