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Mirrors > Home > ILE Home > Th. List > mpoeq123 | Unicode version |
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
mpoeq123 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1526 | . . . 4 | |
2 | nfra1 2506 | . . . 4 | |
3 | 1, 2 | nfan 1563 | . . 3 |
4 | nfv 1526 | . . . 4 | |
5 | nfcv 2317 | . . . . 5 | |
6 | nfv 1526 | . . . . . 6 | |
7 | nfra1 2506 | . . . . . 6 | |
8 | 6, 7 | nfan 1563 | . . . . 5 |
9 | 5, 8 | nfralxy 2513 | . . . 4 |
10 | 4, 9 | nfan 1563 | . . 3 |
11 | nfv 1526 | . . 3 | |
12 | rsp 2522 | . . . . . . 7 | |
13 | rsp 2522 | . . . . . . . . . 10 | |
14 | eqeq2 2185 | . . . . . . . . . 10 | |
15 | 13, 14 | syl6 33 | . . . . . . . . 9 |
16 | 15 | pm5.32d 450 | . . . . . . . 8 |
17 | eleq2 2239 | . . . . . . . . 9 | |
18 | 17 | anbi1d 465 | . . . . . . . 8 |
19 | 16, 18 | sylan9bbr 463 | . . . . . . 7 |
20 | 12, 19 | syl6 33 | . . . . . 6 |
21 | 20 | pm5.32d 450 | . . . . 5 |
22 | eleq2 2239 | . . . . . 6 | |
23 | 22 | anbi1d 465 | . . . . 5 |
24 | 21, 23 | sylan9bbr 463 | . . . 4 |
25 | anass 401 | . . . 4 | |
26 | anass 401 | . . . 4 | |
27 | 24, 25, 26 | 3bitr4g 223 | . . 3 |
28 | 3, 10, 11, 27 | oprabbid 5918 | . 2 |
29 | df-mpo 5870 | . 2 | |
30 | df-mpo 5870 | . 2 | |
31 | 28, 29, 30 | 3eqtr4g 2233 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wceq 1353 wcel 2146 wral 2453 coprab 5866 cmpo 5867 |
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-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-11 1504 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-ext 2157 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1459 df-sb 1761 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-oprab 5869 df-mpo 5870 |
This theorem is referenced by: mpoeq12 5925 mapxpen 6838 |
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