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Mirrors > Home > MPE Home > Th. List > suppssOLD | Structured version Unicode version |
Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppss 6816 as of 28-May-2019. (New usage is discouraged.) |
Ref | Expression |
---|---|
suppssOLD.f |
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suppssOLD.n |
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Ref | Expression |
---|---|
suppssOLD |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppssOLD.f |
. . . 4
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2 | ffn 5654 |
. . . 4
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3 | elpreima 5919 |
. . . 4
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4 | 1, 2, 3 | 3syl 20 |
. . 3
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5 | fvex 5796 |
. . . . . 6
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6 | eldifsn 4095 |
. . . . . 6
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7 | 5, 6 | mpbiran 909 |
. . . . 5
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8 | eldif 3433 |
. . . . . . . 8
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9 | suppssOLD.n |
. . . . . . . 8
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10 | 8, 9 | sylan2br 476 |
. . . . . . 7
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11 | 10 | expr 615 |
. . . . . 6
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12 | 11 | necon1ad 2662 |
. . . . 5
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13 | 7, 12 | syl5bi 217 |
. . . 4
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14 | 13 | expimpd 603 |
. . 3
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15 | 4, 14 | sylbid 215 |
. 2
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16 | 15 | ssrdv 3457 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4508 ax-nul 4516 ax-pr 4626 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2599 df-ne 2644 df-ral 2798 df-rex 2799 df-rab 2802 df-v 3067 df-sbc 3282 df-dif 3426 df-un 3428 df-in 3430 df-ss 3437 df-nul 3733 df-if 3887 df-sn 3973 df-pr 3975 df-op 3979 df-uni 4187 df-br 4388 df-opab 4446 df-id 4731 df-xp 4941 df-rel 4942 df-cnv 4943 df-co 4944 df-dm 4945 df-rn 4946 df-res 4947 df-ima 4948 df-iota 5476 df-fun 5515 df-fn 5516 df-f 5517 df-fv 5521 |
This theorem is referenced by: cantnfp1lem1OLD 8010 cantnfp1lem3OLD 8012 gsumzaddlemOLD 16511 gsumzmhmOLD 16533 gsumzinvOLD 16545 gsumsubOLD 16550 lcomfsupOLD 17087 psrbaglesuppOLD 17539 psrlidmOLD 17578 psrridmOLD 17580 mplsubglemOLD 17616 mpllsslemOLD 17617 mplsubrglemOLD 17622 frlmssuvc1OLD 18327 frlmsslspOLD 18330 |
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