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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 2700 |
. . . . . . . 8
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3 | 1, 2 | syl 14 |
. . . . . . 7
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4 | 3 | adantr 274 |
. . . . . 6
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5 | eldifsni 3660 |
. . . . . . . 8
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6 | oveq2 5790 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | eqeq1d 2149 |
. . . . . . . . . . 11
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8 | suppssov1.o |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | ralrimiva 2508 |
. . . . . . . . . . . 12
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10 | 9 | adantr 274 |
. . . . . . . . . . 11
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11 | suppssov1.b |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 7, 10, 11 | rspcdva 2798 |
. . . . . . . . . 10
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13 | oveq1 5789 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | eqeq1d 2149 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 12, 14 | syl5ibrcom 156 |
. . . . . . . . 9
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16 | 15 | necon3d 2353 |
. . . . . . . 8
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17 | 5, 16 | syl5 32 |
. . . . . . 7
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18 | 17 | imp 123 |
. . . . . 6
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19 | eldifsn 3658 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 4, 18, 19 | sylanbrc 414 |
. . . . 5
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21 | 20 | ex 114 |
. . . 4
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22 | 21 | ss2rabdv 3183 |
. . 3
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23 | eqid 2140 |
. . . 4
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24 | 23 | mptpreima 5040 |
. . 3
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25 | eqid 2140 |
. . . 4
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26 | 25 | mptpreima 5040 |
. . 3
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27 | 22, 24, 26 | 3sstr4g 3145 |
. 2
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28 | suppssov1.s |
. 2
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29 | 27, 28 | sstrd 3112 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-xp 4553 df-rel 4554 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fv 5139 df-ov 5785 |
This theorem is referenced by: suppssof1 6007 |
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