Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elmpocl | Unicode version |
Description: If a two-parameter class is inhabited, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
elmpocl.f |
Ref | Expression |
---|---|
elmpocl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmpocl.f | . . . . . 6 | |
2 | df-mpo 5779 | . . . . . 6 | |
3 | 1, 2 | eqtri 2160 | . . . . 5 |
4 | 3 | dmeqi 4740 | . . . 4 |
5 | dmoprabss 5853 | . . . 4 | |
6 | 4, 5 | eqsstri 3129 | . . 3 |
7 | 1 | mpofun 5873 | . . . . . 6 |
8 | funrel 5140 | . . . . . 6 | |
9 | 7, 8 | ax-mp 5 | . . . . 5 |
10 | relelfvdm 5453 | . . . . 5 | |
11 | 9, 10 | mpan 420 | . . . 4 |
12 | df-ov 5777 | . . . 4 | |
13 | 11, 12 | eleq2s 2234 | . . 3 |
14 | 6, 13 | sseldi 3095 | . 2 |
15 | opelxp 4569 | . 2 | |
16 | 14, 15 | sylib 121 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3530 cxp 4537 cdm 4539 wrel 4544 wfun 5117 cfv 5123 (class class class)co 5774 coprab 5775 cmpo 5776 |
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-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 |
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-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-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 |
This theorem is referenced by: elmpocl1 5969 elmpocl2 5970 elovmpo 5971 elpmi 6561 elmapex 6563 pmsspw 6577 ixxssxr 9683 elixx3g 9684 ixxssixx 9685 eliooxr 9710 elfz2 9797 restsspw 12130 restrcl 12336 ssrest 12351 iscn2 12369 ishmeo 12473 limcrcl 12796 |
Copyright terms: Public domain | W3C validator |