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Mirrors > Home > ILE Home > Th. List > mpteq12f | Unicode version |
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq12f |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 1528 | . . . 4 | |
2 | nfra1 2495 | . . . 4 | |
3 | 1, 2 | nfan 1552 | . . 3 |
4 | nfv 1515 | . . 3 | |
5 | rsp 2511 | . . . . . . 7 | |
6 | 5 | imp 123 | . . . . . 6 |
7 | 6 | eqeq2d 2176 | . . . . 5 |
8 | 7 | pm5.32da 448 | . . . 4 |
9 | sp 1498 | . . . . . 6 | |
10 | 9 | eleq2d 2234 | . . . . 5 |
11 | 10 | anbi1d 461 | . . . 4 |
12 | 8, 11 | sylan9bbr 459 | . . 3 |
13 | 3, 4, 12 | opabbid 4041 | . 2 |
14 | df-mpt 4039 | . 2 | |
15 | df-mpt 4039 | . 2 | |
16 | 13, 14, 15 | 3eqtr4g 2222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1340 wceq 1342 wcel 2135 wral 2442 copab 4036 cmpt 4037 |
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-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-ral 2447 df-opab 4038 df-mpt 4039 |
This theorem is referenced by: mpteq12dva 4057 mpteq12 4059 mpteq2ia 4062 mpteq2da 4065 |
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