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Mirrors > Home > ILE Home > Th. List > suppssfv | Unicode version |
Description: Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
Ref | Expression |
---|---|
suppssfv.a | |
suppssfv.f | |
suppssfv.v |
Ref | Expression |
---|---|
suppssfv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsni 3622 | . . . . 5 | |
2 | suppssfv.v | . . . . . . . . 9 | |
3 | elex 2671 | . . . . . . . . 9 | |
4 | 2, 3 | syl 14 | . . . . . . . 8 |
5 | 4 | adantr 274 | . . . . . . 7 |
6 | suppssfv.f | . . . . . . . . . . 11 | |
7 | fveq2 5389 | . . . . . . . . . . . 12 | |
8 | 7 | eqeq1d 2126 | . . . . . . . . . . 11 |
9 | 6, 8 | syl5ibrcom 156 | . . . . . . . . . 10 |
10 | 9 | necon3d 2329 | . . . . . . . . 9 |
11 | 10 | adantr 274 | . . . . . . . 8 |
12 | 11 | imp 123 | . . . . . . 7 |
13 | eldifsn 3620 | . . . . . . 7 | |
14 | 5, 12, 13 | sylanbrc 413 | . . . . . 6 |
15 | 14 | ex 114 | . . . . 5 |
16 | 1, 15 | syl5 32 | . . . 4 |
17 | 16 | ss2rabdv 3148 | . . 3 |
18 | eqid 2117 | . . . 4 | |
19 | 18 | mptpreima 5002 | . . 3 |
20 | eqid 2117 | . . . 4 | |
21 | 20 | mptpreima 5002 | . . 3 |
22 | 17, 19, 21 | 3sstr4g 3110 | . 2 |
23 | suppssfv.a | . 2 | |
24 | 22, 23 | sstrd 3077 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wne 2285 crab 2397 cvv 2660 cdif 3038 wss 3041 csn 3497 cmpt 3959 ccnv 4508 cima 4512 cfv 5093 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-xp 4515 df-rel 4516 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fv 5101 |
This theorem is referenced by: (None) |
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