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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 2763 |
. . . . . . . 8
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3 | 1, 2 | syl 14 |
. . . . . . 7
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4 | 3 | adantr 276 |
. . . . . 6
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5 | eldifsni 3736 |
. . . . . . . 8
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6 | oveq2 5905 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | eqeq1d 2198 |
. . . . . . . . . . 11
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8 | suppssov1.o |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | ralrimiva 2563 |
. . . . . . . . . . . 12
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10 | 9 | adantr 276 |
. . . . . . . . . . 11
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11 | suppssov1.b |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 7, 10, 11 | rspcdva 2861 |
. . . . . . . . . 10
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13 | oveq1 5904 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | eqeq1d 2198 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 12, 14 | syl5ibrcom 157 |
. . . . . . . . 9
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16 | 15 | necon3d 2404 |
. . . . . . . 8
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17 | 5, 16 | syl5 32 |
. . . . . . 7
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18 | 17 | imp 124 |
. . . . . 6
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19 | eldifsn 3734 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 4, 18, 19 | sylanbrc 417 |
. . . . 5
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21 | 20 | ex 115 |
. . . 4
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22 | 21 | ss2rabdv 3251 |
. . 3
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23 | eqid 2189 |
. . . 4
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24 | 23 | mptpreima 5140 |
. . 3
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25 | eqid 2189 |
. . . 4
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26 | 25 | mptpreima 5140 |
. . 3
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27 | 22, 24, 26 | 3sstr4g 3213 |
. 2
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28 | suppssov1.s |
. 2
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29 | 27, 28 | sstrd 3180 |
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 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-xp 4650 df-rel 4651 df-cnv 4652 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fv 5243 df-ov 5900 |
This theorem is referenced by: suppssof1 6125 |
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