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Mirrors > Home > ILE Home > Th. List > relelfvdm | Unicode version |
Description: If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.) |
Ref | Expression |
---|---|
relelfvdm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfv 5176 | . . . . . 6 | |
2 | exsimpr 1509 | . . . . . 6 | |
3 | 1, 2 | sylbi 114 | . . . . 5 |
4 | equsb1 1668 | . . . . . . . 8 | |
5 | spsbbi 1725 | . . . . . . . 8 | |
6 | 4, 5 | mpbiri 157 | . . . . . . 7 |
7 | nfv 1421 | . . . . . . . 8 | |
8 | breq2 3768 | . . . . . . . 8 | |
9 | 7, 8 | sbie 1674 | . . . . . . 7 |
10 | 6, 9 | sylib 127 | . . . . . 6 |
11 | 10 | eximi 1491 | . . . . 5 |
12 | 3, 11 | syl 14 | . . . 4 |
13 | 12 | anim2i 324 | . . 3 |
14 | 19.42v 1786 | . . 3 | |
15 | 13, 14 | sylibr 137 | . 2 |
16 | releldm 4569 | . . 3 | |
17 | 16 | exlimiv 1489 | . 2 |
18 | 15, 17 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wex 1381 wcel 1393 wsb 1645 class class class wbr 3764 cdm 4345 wrel 4350 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-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 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 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-xp 4351 df-rel 4352 df-dm 4355 df-iota 4867 df-fv 4910 |
This theorem is referenced by: elmpt2cl 5698 mpt2xopn0yelv 5854 eluzel2 8478 |
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