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Mirrors > Home > ILE Home > Th. List > fnresdisj | Unicode version |
Description: A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.) |
Ref | Expression |
---|---|
fnresdisj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4754 |
. . 3
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2 | reldm0 4667 |
. . 3
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3 | 1, 2 | ax-mp 7 |
. 2
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4 | dmres 4747 |
. . . . 5
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5 | incom 3193 |
. . . . 5
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6 | 4, 5 | eqtri 2109 |
. . . 4
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7 | fndm 5126 |
. . . . 5
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8 | 7 | ineq1d 3201 |
. . . 4
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9 | 6, 8 | syl5eq 2133 |
. . 3
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10 | 9 | eqeq1d 2097 |
. 2
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11 | 3, 10 | syl5rbb 192 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-xp 4458 df-rel 4459 df-dm 4462 df-res 4464 df-fn 5031 |
This theorem is referenced by: fvsnun2 5509 fseq1p1m1 9569 |
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