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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 2604 | . . . . . 6 | |
2 | 1 | snex 3957 | . . . . 5 |
3 | 2 | rnex 4617 | . . . 4 |
4 | 3 | uniex 4192 | . . 3 |
5 | df-2nd 5788 | . . 3 | |
6 | 4, 5 | fnmpti 5047 | . 2 |
7 | 5 | rnmpt 4600 | . . 3 |
8 | vex 2604 | . . . . 5 | |
9 | 8, 8 | opex 3984 | . . . . . 6 |
10 | 8, 8 | op2nda 4825 | . . . . . . 7 |
11 | 10 | eqcomi 2085 | . . . . . 6 |
12 | sneq 3409 | . . . . . . . . . 10 | |
13 | 12 | rneqd 4581 | . . . . . . . . 9 |
14 | 13 | unieqd 3612 | . . . . . . . 8 |
15 | 14 | eqeq2d 2092 | . . . . . . 7 |
16 | 15 | rspcev 2701 | . . . . . 6 |
17 | 9, 11, 16 | mp2an 416 | . . . . 5 |
18 | 8, 17 | 2th 172 | . . . 4 |
19 | 18 | abbi2i 2193 | . . 3 |
20 | 7, 19 | eqtr4i 2104 | . 2 |
21 | df-fo 4928 | . 2 | |
22 | 6, 20, 21 | mpbir2an 883 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1284 wcel 1433 cab 2067 wrex 2349 cvv 2601 csn 3398 cop 3401 cuni 3601 crn 4364 wfn 4917 wfo 4920 c2nd 5786 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-fun 4924 df-fn 4925 df-fo 4928 df-2nd 5788 |
This theorem is referenced by: 2ndcof 5811 2ndexg 5815 df2nd2 5861 2ndconst 5863 |
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