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Mirrors > Home > ILE Home > Th. List > fo2nd | Unicode version |
Description: The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
fo2nd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . . . 6 | |
2 | snexgOLD 3935 | . . . . . 6 | |
3 | 1, 2 | ax-mp 7 | . . . . 5 |
4 | 3 | rnex 4599 | . . . 4 |
5 | 4 | uniex 4174 | . . 3 |
6 | df-2nd 5768 | . . 3 | |
7 | 5, 6 | fnmpti 5027 | . 2 |
8 | 6 | rnmpt 4582 | . . 3 |
9 | vex 2560 | . . . . 5 | |
10 | 9, 9 | opex 3966 | . . . . . 6 |
11 | 9, 9 | op2nda 4805 | . . . . . . 7 |
12 | 11 | eqcomi 2044 | . . . . . 6 |
13 | sneq 3386 | . . . . . . . . . 10 | |
14 | 13 | rneqd 4563 | . . . . . . . . 9 |
15 | 14 | unieqd 3591 | . . . . . . . 8 |
16 | 15 | eqeq2d 2051 | . . . . . . 7 |
17 | 16 | rspcev 2656 | . . . . . 6 |
18 | 10, 12, 17 | mp2an 402 | . . . . 5 |
19 | 9, 18 | 2th 163 | . . . 4 |
20 | 19 | abbi2i 2152 | . . 3 |
21 | 8, 20 | eqtr4i 2063 | . 2 |
22 | df-fo 4908 | . 2 | |
23 | 7, 21, 22 | mpbir2an 849 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1243 wcel 1393 cab 2026 wrex 2307 cvv 2557 csn 3375 cop 3378 cuni 3580 crn 4346 wfn 4897 wfo 4900 c2nd 5766 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-fo 4908 df-2nd 5768 |
This theorem is referenced by: 2ndcof 5791 2ndexg 5795 df2nd2 5841 2ndconst 5843 |
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