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Mirrors > Home > ILE Home > Th. List > mpoxopn0yelv | Unicode version |
Description: If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
Ref | Expression |
---|---|
mpoxopn0yelv.f |
Ref | Expression |
---|---|
mpoxopn0yelv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoxopn0yelv.f | . . . . 5 | |
2 | 1 | dmmpossx 6097 | . . . 4 |
3 | 1 | mpofun 5873 | . . . . . . 7 |
4 | funrel 5140 | . . . . . . 7 | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 |
6 | relelfvdm 5453 | . . . . . 6 | |
7 | 5, 6 | mpan 420 | . . . . 5 |
8 | df-ov 5777 | . . . . 5 | |
9 | 7, 8 | eleq2s 2234 | . . . 4 |
10 | 2, 9 | sseldi 3095 | . . 3 |
11 | fveq2 5421 | . . . . 5 | |
12 | 11 | opeliunxp2 4679 | . . . 4 |
13 | 12 | simprbi 273 | . . 3 |
14 | 10, 13 | syl 14 | . 2 |
15 | op1stg 6048 | . . 3 | |
16 | 15 | eleq2d 2209 | . 2 |
17 | 14, 16 | syl5ib 153 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 csn 3527 cop 3530 ciun 3813 cxp 4537 cdm 4539 wrel 4544 wfun 5117 cfv 5123 (class class class)co 5774 cmpo 5776 c1st 6036 |
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-sbc 2910 df-csb 3004 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-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: mpoxopovel 6138 |
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