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Mirrors > Home > ILE Home > Th. List > fo1st | 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 |
---|---|
fo1st |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2663 | . . . . . 6 | |
2 | 1 | snex 4079 | . . . . 5 |
3 | 2 | dmex 4775 | . . . 4 |
4 | 3 | uniex 4329 | . . 3 |
5 | df-1st 6006 | . . 3 | |
6 | 4, 5 | fnmpti 5221 | . 2 |
7 | 5 | rnmpt 4757 | . . 3 |
8 | vex 2663 | . . . . 5 | |
9 | 8, 8 | opex 4121 | . . . . . 6 |
10 | 8, 8 | op1sta 4990 | . . . . . . 7 |
11 | 10 | eqcomi 2121 | . . . . . 6 |
12 | sneq 3508 | . . . . . . . . . 10 | |
13 | 12 | dmeqd 4711 | . . . . . . . . 9 |
14 | 13 | unieqd 3717 | . . . . . . . 8 |
15 | 14 | eqeq2d 2129 | . . . . . . 7 |
16 | 15 | rspcev 2763 | . . . . . 6 |
17 | 9, 11, 16 | mp2an 422 | . . . . 5 |
18 | 8, 17 | 2th 173 | . . . 4 |
19 | 18 | abbi2i 2232 | . . 3 |
20 | 7, 19 | eqtr4i 2141 | . 2 |
21 | df-fo 5099 | . 2 | |
22 | 6, 20, 21 | mpbir2an 911 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1316 wcel 1465 cab 2103 wrex 2394 cvv 2660 csn 3497 cop 3500 cuni 3706 cdm 4509 crn 4510 wfn 5088 wfo 5091 c1st 6004 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-fo 5099 df-1st 6006 |
This theorem is referenced by: 1stcof 6029 1stexg 6033 df1st2 6084 1stconst 6086 algrflem 6094 algrflemg 6095 suplocexprlemell 7489 suplocexprlem2b 7490 suplocexprlemlub 7500 upxp 12368 uptx 12370 cnmpt1st 12384 |
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