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Mirrors > Home > ILE Home > Th. List > fssres | Unicode version |
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.) |
Ref | Expression |
---|---|
fssres |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 5135 |
. . 3
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2 | fnssres 5244 |
. . . . 5
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3 | resss 4851 |
. . . . . . 7
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4 | rnss 4777 |
. . . . . . 7
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5 | 3, 4 | ax-mp 5 |
. . . . . 6
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6 | sstr 3110 |
. . . . . 6
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7 | 5, 6 | mpan 421 |
. . . . 5
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8 | 2, 7 | anim12i 336 |
. . . 4
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9 | 8 | an32s 558 |
. . 3
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10 | 1, 9 | sylanb 282 |
. 2
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11 | df-f 5135 |
. 2
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12 | 10, 11 | sylibr 133 |
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 |
This theorem is referenced by: fssresd 5307 fssres2 5308 fresin 5309 f1ssres 5345 feqresmpt 5483 f2ndf 6131 elmapssres 6575 pmresg 6578 finomni 7020 fseq1p1m1 9905 hmeores 12523 limcdifap 12839 012of 13363 2o01f 13364 |
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