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Mirrors > Home > ILE Home > Th. List > mpofun | Unicode version |
Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.) |
Ref | Expression |
---|---|
mpofun.1 |
Ref | Expression |
---|---|
mpofun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr3 2177 | . . . . . 6 | |
2 | 1 | ad2ant2l 500 | . . . . 5 |
3 | 2 | gen2 1430 | . . . 4 |
4 | eqeq1 2164 | . . . . . 6 | |
5 | 4 | anbi2d 460 | . . . . 5 |
6 | 5 | mo4 2067 | . . . 4 |
7 | 3, 6 | mpbir 145 | . . 3 |
8 | 7 | funoprab 5923 | . 2 |
9 | mpofun.1 | . . . 4 | |
10 | df-mpo 5831 | . . . 4 | |
11 | 9, 10 | eqtri 2178 | . . 3 |
12 | 11 | funeqi 5193 | . 2 |
13 | 8, 12 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1333 wceq 1335 wmo 2007 wcel 2128 wfun 5166 coprab 5827 cmpo 5828 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-fun 5174 df-oprab 5830 df-mpo 5831 |
This theorem is referenced by: elmpocl 6020 ofexg 6038 mpoexxg 6160 mpoxopn0yelv 6188 |
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