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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 3647 | . . . . 5 | |
2 | suppssfv.v | . . . . . . . . 9 | |
3 | elex 2692 | . . . . . . . . 9 | |
4 | 2, 3 | syl 14 | . . . . . . . 8 |
5 | 4 | adantr 274 | . . . . . . 7 |
6 | suppssfv.f | . . . . . . . . . . 11 | |
7 | fveq2 5414 | . . . . . . . . . . . 12 | |
8 | 7 | eqeq1d 2146 | . . . . . . . . . . 11 |
9 | 6, 8 | syl5ibrcom 156 | . . . . . . . . . 10 |
10 | 9 | necon3d 2350 | . . . . . . . . 9 |
11 | 10 | adantr 274 | . . . . . . . 8 |
12 | 11 | imp 123 | . . . . . . 7 |
13 | eldifsn 3645 | . . . . . . 7 | |
14 | 5, 12, 13 | sylanbrc 413 | . . . . . 6 |
15 | 14 | ex 114 | . . . . 5 |
16 | 1, 15 | syl5 32 | . . . 4 |
17 | 16 | ss2rabdv 3173 | . . 3 |
18 | eqid 2137 | . . . 4 | |
19 | 18 | mptpreima 5027 | . . 3 |
20 | eqid 2137 | . . . 4 | |
21 | 20 | mptpreima 5027 | . . 3 |
22 | 17, 19, 21 | 3sstr4g 3135 | . 2 |
23 | suppssfv.a | . 2 | |
24 | 22, 23 | sstrd 3102 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wne 2306 crab 2418 cvv 2681 cdif 3063 wss 3066 csn 3522 cmpt 3984 ccnv 4533 cima 4537 cfv 5118 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fv 5126 |
This theorem is referenced by: (None) |
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