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Mirrors > Home > ILE Home > Th. List > f1ssres | Unicode version |
Description: A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.) |
Ref | Expression |
---|---|
f1ssres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 5328 | . . 3 | |
2 | fssres 5298 | . . 3 | |
3 | 1, 2 | sylan 281 | . 2 |
4 | df-f1 5128 | . . . . 5 | |
5 | 4 | simprbi 273 | . . . 4 |
6 | funres11 5195 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 7 | adantr 274 | . 2 |
9 | df-f1 5128 | . 2 | |
10 | 3, 8, 9 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wss 3071 ccnv 4538 cres 4541 wfun 5117 wf 5119 wf1 5120 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 |
This theorem is referenced by: f1resf1 5338 f1ores 5382 |
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