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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 5336 |
. . 3
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2 | fssres 5306 |
. . 3
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3 | 1, 2 | sylan 281 |
. 2
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4 | df-f1 5136 |
. . . . 5
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5 | 4 | simprbi 273 |
. . . 4
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6 | funres11 5203 |
. . . 4
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7 | 5, 6 | syl 14 |
. . 3
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8 | 7 | adantr 274 |
. 2
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9 | df-f1 5136 |
. 2
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10 | 3, 8, 9 | sylanbrc 414 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 |
This theorem is referenced by: f1resf1 5346 f1ores 5390 |
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