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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 4950 |
. . 3
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2 | reldm0 4860 |
. . 3
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3 | 1, 2 | ax-mp 5 |
. 2
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4 | dmres 4943 |
. . . . 5
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5 | incom 3342 |
. . . . 5
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6 | 4, 5 | eqtri 2210 |
. . . 4
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7 | fndm 5330 |
. . . . 5
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8 | 7 | ineq1d 3350 |
. . . 4
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9 | 6, 8 | eqtrid 2234 |
. . 3
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10 | 9 | eqeq1d 2198 |
. 2
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11 | 3, 10 | bitr2id 193 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4647 df-rel 4648 df-dm 4651 df-res 4653 df-fn 5234 |
This theorem is referenced by: fvsnun2 5730 fseq1p1m1 10112 |
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