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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 1506 | . . . 4 | |
2 | nfra1 2443 | . . . 4 | |
3 | 1, 2 | nfan 1529 | . . 3 |
4 | nfv 1493 | . . 3 | |
5 | rsp 2457 | . . . . . . 7 | |
6 | 5 | imp 123 | . . . . . 6 |
7 | 6 | eqeq2d 2129 | . . . . 5 |
8 | 7 | pm5.32da 447 | . . . 4 |
9 | sp 1473 | . . . . . 6 | |
10 | 9 | eleq2d 2187 | . . . . 5 |
11 | 10 | anbi1d 460 | . . . 4 |
12 | 8, 11 | sylan9bbr 458 | . . 3 |
13 | 3, 4, 12 | opabbid 3963 | . 2 |
14 | df-mpt 3961 | . 2 | |
15 | df-mpt 3961 | . 2 | |
16 | 13, 14, 15 | 3eqtr4g 2175 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1314 wceq 1316 wcel 1465 wral 2393 copab 3958 cmpt 3959 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-ral 2398 df-opab 3960 df-mpt 3961 |
This theorem is referenced by: mpteq12dva 3979 mpteq12 3981 mpteq2ia 3984 mpteq2da 3987 |
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