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Mirrors > Home > ILE Home > Th. List > suppssov1 | Unicode version |
Description: Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
Ref | Expression |
---|---|
suppssov1.s |
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suppssov1.o |
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suppssov1.a |
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suppssov1.b |
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Ref | Expression |
---|---|
suppssov1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppssov1.a |
. . . . . . . 8
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2 | elex 2630 |
. . . . . . . 8
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3 | 1, 2 | syl 14 |
. . . . . . 7
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4 | 3 | adantr 270 |
. . . . . 6
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5 | eldifsni 3569 |
. . . . . . . 8
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6 | oveq2 5660 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | eqeq1d 2096 |
. . . . . . . . . . 11
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8 | suppssov1.o |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | ralrimiva 2446 |
. . . . . . . . . . . 12
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10 | 9 | adantr 270 |
. . . . . . . . . . 11
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11 | suppssov1.b |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 7, 10, 11 | rspcdva 2727 |
. . . . . . . . . 10
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13 | oveq1 5659 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | eqeq1d 2096 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 12, 14 | syl5ibrcom 155 |
. . . . . . . . 9
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16 | 15 | necon3d 2299 |
. . . . . . . 8
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17 | 5, 16 | syl5 32 |
. . . . . . 7
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18 | 17 | imp 122 |
. . . . . 6
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19 | eldifsn 3567 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 4, 18, 19 | sylanbrc 408 |
. . . . 5
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21 | 20 | ex 113 |
. . . 4
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22 | 21 | ss2rabdv 3102 |
. . 3
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23 | eqid 2088 |
. . . 4
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24 | 23 | mptpreima 4924 |
. . 3
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25 | eqid 2088 |
. . . 4
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26 | 25 | mptpreima 4924 |
. . 3
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27 | 22, 24, 26 | 3sstr4g 3067 |
. 2
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28 | suppssov1.s |
. 2
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29 | 27, 28 | sstrd 3035 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-xp 4444 df-rel 4445 df-cnv 4446 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fv 5023 df-ov 5655 |
This theorem is referenced by: suppssof1 5872 |
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