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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 2724 | . . . . . . . 8 | |
2 | 1 | a1i 9 | . . . . . . 7 Se |
3 | simp3 988 | . . . . . . . 8 Se | |
4 | simp2 987 | . . . . . . . . 9 Se | |
5 | brcnvg 4779 | . . . . . . . . 9 | |
6 | 1, 4, 5 | sylancr 411 | . . . . . . . 8 Se |
7 | 3, 6 | mpbird 166 | . . . . . . 7 Se |
8 | breq1 3979 | . . . . . . . 8 | |
9 | 8 | elrab 2877 | . . . . . . 7 |
10 | 2, 7, 9 | sylanbrc 414 | . . . . . 6 Se |
11 | elssuni 3811 | . . . . . 6 | |
12 | 10, 11 | syl 14 | . . . . 5 Se |
13 | 12 | 3expia 1194 | . . . 4 Se |
14 | 13 | alrimiv 1861 | . . 3 Se |
15 | fvss 5494 | . . 3 | |
16 | 14, 15 | syl 14 | . 2 Se |
17 | seex 4307 | . . 3 Se | |
18 | uniexg 4411 | . . 3 | |
19 | 17, 18 | syl 14 | . 2 Se |
20 | ssexg 4115 | . 2 | |
21 | 16, 19, 20 | syl2anc 409 | 1 Se |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wal 1340 wcel 2135 crab 2446 cvv 2721 wss 3111 cuni 3783 class class class wbr 3976 Se wse 4301 ccnv 4597 cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-se 4305 df-cnv 4606 df-iota 5147 df-fv 5190 |
This theorem is referenced by: (None) |
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