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Mirrors > Home > ILE Home > Th. List > mpt2eq123 | 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 |
---|---|
mpt2eq123 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . . . 4 | |
2 | nfra1 2355 | . . . 4 | |
3 | 1, 2 | nfan 1457 | . . 3 |
4 | nfv 1421 | . . . 4 | |
5 | nfcv 2178 | . . . . 5 | |
6 | nfv 1421 | . . . . . 6 | |
7 | nfra1 2355 | . . . . . 6 | |
8 | 6, 7 | nfan 1457 | . . . . 5 |
9 | 5, 8 | nfralxy 2360 | . . . 4 |
10 | 4, 9 | nfan 1457 | . . 3 |
11 | nfv 1421 | . . 3 | |
12 | rsp 2369 | . . . . . . 7 | |
13 | rsp 2369 | . . . . . . . . . 10 | |
14 | eqeq2 2049 | . . . . . . . . . 10 | |
15 | 13, 14 | syl6 29 | . . . . . . . . 9 |
16 | 15 | pm5.32d 423 | . . . . . . . 8 |
17 | eleq2 2101 | . . . . . . . . 9 | |
18 | 17 | anbi1d 438 | . . . . . . . 8 |
19 | 16, 18 | sylan9bbr 436 | . . . . . . 7 |
20 | 12, 19 | syl6 29 | . . . . . 6 |
21 | 20 | pm5.32d 423 | . . . . 5 |
22 | eleq2 2101 | . . . . . 6 | |
23 | 22 | anbi1d 438 | . . . . 5 |
24 | 21, 23 | sylan9bbr 436 | . . . 4 |
25 | anass 381 | . . . 4 | |
26 | anass 381 | . . . 4 | |
27 | 24, 25, 26 | 3bitr4g 212 | . . 3 |
28 | 3, 10, 11, 27 | oprabbid 5558 | . 2 |
29 | df-mpt2 5517 | . 2 | |
30 | df-mpt2 5517 | . 2 | |
31 | 28, 29, 30 | 3eqtr4g 2097 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 coprab 5513 cmpt2 5514 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-oprab 5516 df-mpt2 5517 |
This theorem is referenced by: mpt2eq12 5565 |
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