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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 |
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Ref | Expression |
---|---|
elmpocl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmpocl.f |
. . . . . 6
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2 | df-mpo 5879 |
. . . . . 6
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3 | 1, 2 | eqtri 2198 |
. . . . 5
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4 | 3 | dmeqi 4828 |
. . . 4
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5 | dmoprabss 5956 |
. . . 4
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6 | 4, 5 | eqsstri 3187 |
. . 3
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7 | 1 | mpofun 5976 |
. . . . . 6
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8 | funrel 5233 |
. . . . . 6
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9 | 7, 8 | ax-mp 5 |
. . . . 5
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10 | relelfvdm 5547 |
. . . . 5
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11 | 9, 10 | mpan 424 |
. . . 4
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12 | df-ov 5877 |
. . . 4
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13 | 11, 12 | eleq2s 2272 |
. . 3
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14 | 6, 13 | sselid 3153 |
. 2
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15 | opelxp 4656 |
. 2
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16 | 14, 15 | sylib 122 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 |
This theorem is referenced by: elmpocl1 6069 elmpocl2 6070 elovmpo 6071 elpmi 6666 elmapex 6668 pmsspw 6682 ixxssxr 9899 elixx3g 9900 ixxssixx 9901 eliooxr 9926 elfz2 10014 restsspw 12697 ismhm 12852 restrcl 13603 ssrest 13618 iscn2 13636 ishmeo 13740 limcrcl 14063 |
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