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Mirrors > Home > MPE Home > Th. List > fo1st | Structured version Visualization version 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 | snex 4908 | . . . . 5 | |
2 | 1 | dmex 7099 | . . . 4 |
3 | 2 | uniex 6953 | . . 3 |
4 | df-1st 7168 | . . 3 | |
5 | 3, 4 | fnmpti 6022 | . 2 |
6 | 4 | rnmpt 5371 | . . 3 |
7 | vex 3203 | . . . . 5 | |
8 | opex 4932 | . . . . . 6 | |
9 | 7, 7 | op1sta 5617 | . . . . . . 7 |
10 | 9 | eqcomi 2631 | . . . . . 6 |
11 | sneq 4187 | . . . . . . . . . 10 | |
12 | 11 | dmeqd 5326 | . . . . . . . . 9 |
13 | 12 | unieqd 4446 | . . . . . . . 8 |
14 | 13 | eqeq2d 2632 | . . . . . . 7 |
15 | 14 | rspcev 3309 | . . . . . 6 |
16 | 8, 10, 15 | mp2an 708 | . . . . 5 |
17 | 7, 16 | 2th 254 | . . . 4 |
18 | 17 | abbi2i 2738 | . . 3 |
19 | 6, 18 | eqtr4i 2647 | . 2 |
20 | df-fo 5894 | . 2 | |
21 | 5, 19, 20 | mpbir2an 955 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cab 2608 wrex 2913 cvv 3200 csn 4177 cop 4183 cuni 4436 cdm 5114 crn 5115 wfn 5883 wfo 5886 c1st 7166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-fo 5894 df-1st 7168 |
This theorem is referenced by: 1stcof 7196 df1st2 7263 1stconst 7265 fsplit 7282 algrflem 7286 fpwwe 9468 axpre-sup 9990 homadm 16690 homacd 16691 dmaf 16699 cdaf 16700 1stf1 16832 1stf2 16833 1stfcl 16837 upxp 21426 uptx 21428 cnmpt1st 21471 bcthlem4 23124 uniiccdif 23346 vafval 27458 smfval 27460 0vfval 27461 vsfval 27488 xppreima 29449 xppreima2 29450 1stpreimas 29483 1stpreima 29484 gsummpt2d 29781 cnre2csqima 29957 br1steq 31670 br1steqg 31672 poimirlem26 33435 poimirlem27 33436 |
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