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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 2663 | . . . . . . . 8 | |
2 | 1 | a1i 9 | . . . . . . 7 Se |
3 | simp3 968 | . . . . . . . 8 Se | |
4 | simp2 967 | . . . . . . . . 9 Se | |
5 | brcnvg 4690 | . . . . . . . . 9 | |
6 | 1, 4, 5 | sylancr 410 | . . . . . . . 8 Se |
7 | 3, 6 | mpbird 166 | . . . . . . 7 Se |
8 | breq1 3902 | . . . . . . . 8 | |
9 | 8 | elrab 2813 | . . . . . . 7 |
10 | 2, 7, 9 | sylanbrc 413 | . . . . . 6 Se |
11 | elssuni 3734 | . . . . . 6 | |
12 | 10, 11 | syl 14 | . . . . 5 Se |
13 | 12 | 3expia 1168 | . . . 4 Se |
14 | 13 | alrimiv 1830 | . . 3 Se |
15 | fvss 5403 | . . 3 | |
16 | 14, 15 | syl 14 | . 2 Se |
17 | seex 4227 | . . 3 Se | |
18 | uniexg 4331 | . . 3 | |
19 | 17, 18 | syl 14 | . 2 Se |
20 | ssexg 4037 | . 2 | |
21 | 16, 19, 20 | syl2anc 408 | 1 Se |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wal 1314 wcel 1465 crab 2397 cvv 2660 wss 3041 cuni 3706 class class class wbr 3899 Se wse 4221 ccnv 4508 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-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-13 1476 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 ax-un 4325 |
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-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 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-se 4225 df-cnv 4517 df-iota 5058 df-fv 5101 |
This theorem is referenced by: (None) |
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