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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 2689 | . . . . . . . 8 | |
2 | 1 | a1i 9 | . . . . . . 7 Se |
3 | simp3 983 | . . . . . . . 8 Se | |
4 | simp2 982 | . . . . . . . . 9 Se | |
5 | brcnvg 4720 | . . . . . . . . 9 | |
6 | 1, 4, 5 | sylancr 410 | . . . . . . . 8 Se |
7 | 3, 6 | mpbird 166 | . . . . . . 7 Se |
8 | breq1 3932 | . . . . . . . 8 | |
9 | 8 | elrab 2840 | . . . . . . 7 |
10 | 2, 7, 9 | sylanbrc 413 | . . . . . 6 Se |
11 | elssuni 3764 | . . . . . 6 | |
12 | 10, 11 | syl 14 | . . . . 5 Se |
13 | 12 | 3expia 1183 | . . . 4 Se |
14 | 13 | alrimiv 1846 | . . 3 Se |
15 | fvss 5435 | . . 3 | |
16 | 14, 15 | syl 14 | . 2 Se |
17 | seex 4257 | . . 3 Se | |
18 | uniexg 4361 | . . 3 | |
19 | 17, 18 | syl 14 | . 2 Se |
20 | ssexg 4067 | . 2 | |
21 | 16, 19, 20 | syl2anc 408 | 1 Se |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wal 1329 wcel 1480 crab 2420 cvv 2686 wss 3071 cuni 3736 class class class wbr 3929 Se wse 4251 ccnv 4538 cfv 5123 |
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-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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-se 4255 df-cnv 4547 df-iota 5088 df-fv 5131 |
This theorem is referenced by: (None) |
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