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Mirrors > Home > ILE Home > Th. List > sefvex | Unicode version |
Description: If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
sefvex | Se |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . . . . . 8 | |
2 | 1 | a1i 9 | . . . . . . 7 Se |
3 | simp3 906 | . . . . . . . 8 Se | |
4 | simp2 905 | . . . . . . . . 9 Se | |
5 | brcnvg 4516 | . . . . . . . . 9 | |
6 | 1, 4, 5 | sylancr 393 | . . . . . . . 8 Se |
7 | 3, 6 | mpbird 156 | . . . . . . 7 Se |
8 | breq1 3767 | . . . . . . . 8 | |
9 | 8 | elrab 2698 | . . . . . . 7 |
10 | 2, 7, 9 | sylanbrc 394 | . . . . . 6 Se |
11 | elssuni 3608 | . . . . . 6 | |
12 | 10, 11 | syl 14 | . . . . 5 Se |
13 | 12 | 3expia 1106 | . . . 4 Se |
14 | 13 | alrimiv 1754 | . . 3 Se |
15 | fvss 5189 | . . 3 | |
16 | 14, 15 | syl 14 | . 2 Se |
17 | seex 4072 | . . 3 Se | |
18 | uniexg 4175 | . . 3 | |
19 | 17, 18 | syl 14 | . 2 Se |
20 | ssexg 3896 | . 2 | |
21 | 16, 19, 20 | syl2anc 391 | 1 Se |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wal 1241 wcel 1393 crab 2310 cvv 2557 wss 2917 cuni 3580 class class class wbr 3764 Se wse 4066 ccnv 4344 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-se 4070 df-cnv 4353 df-iota 4867 df-fv 4910 |
This theorem is referenced by: (None) |
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