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Mirrors > Home > ILE Home > Th. List > elpm2r | Unicode version |
Description: Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.) |
Ref | Expression |
---|---|
elpm2r |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 5366 |
. . . . . . 7
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2 | 1 | feq2d 5348 |
. . . . . 6
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3 | 1 | sseq1d 3184 |
. . . . . 6
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4 | 2, 3 | anbi12d 473 |
. . . . 5
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5 | 4 | adantr 276 |
. . . 4
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6 | 5 | ibir 177 |
. . 3
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7 | elpm2g 6658 |
. . 3
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8 | 6, 7 | syl5ibr 156 |
. 2
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9 | 8 | imp 124 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pm 6644 |
This theorem is referenced by: fpmg 6667 pmresg 6669 ennnfonelemg 12374 lmbrf 13348 ellimc3apf 13762 dvfvalap 13783 dvmulxxbr 13799 dvaddxx 13800 dvmulxx 13801 dviaddf 13802 dvimulf 13803 dvcoapbr 13804 dvmptclx 13813 |
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