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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 5435 |
. . 3
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2 | fssres 5405 |
. . 3
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3 | 1, 2 | sylan 283 |
. 2
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4 | df-f1 5235 |
. . . . 5
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5 | 4 | simprbi 275 |
. . . 4
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6 | funres11 5302 |
. . . 4
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7 | 5, 6 | syl 14 |
. . 3
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8 | 7 | adantr 276 |
. 2
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9 | df-f1 5235 |
. 2
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10 | 3, 8, 9 | sylanbrc 417 |
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 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ral 2472 df-rex 2473 df-v 2753 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-br 4018 df-opab 4079 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 |
This theorem is referenced by: f1resf1 5445 f1ores 5490 conjsubgen 13177 |
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