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Mirrors > Home > MPE Home > Th. List > fo2nd | 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 |
---|---|
fo2nd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 4908 | . . . . 5 | |
2 | 1 | rnex 7100 | . . . 4 |
3 | 2 | uniex 6953 | . . 3 |
4 | df-2nd 7169 | . . 3 | |
5 | 3, 4 | fnmpti 6022 | . 2 |
6 | 4 | rnmpt 5371 | . . 3 |
7 | vex 3203 | . . . . 5 | |
8 | opex 4932 | . . . . . 6 | |
9 | 7, 7 | op2nda 5620 | . . . . . . 7 |
10 | 9 | eqcomi 2631 | . . . . . 6 |
11 | sneq 4187 | . . . . . . . . . 10 | |
12 | 11 | rneqd 5353 | . . . . . . . . 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 crn 5115 wfn 5883 wfo 5886 c2nd 7167 |
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-2nd 7169 |
This theorem is referenced by: 2ndcof 7197 df2nd2 7264 2ndconst 7266 iunfo 9361 cdaf 16700 2ndf1 16835 2ndf2 16836 2ndfcl 16838 gsum2dlem2 18370 upxp 21426 uptx 21428 cnmpt2nd 21472 uniiccdif 23346 xppreima 29449 xppreima2 29450 2ndpreima 29485 gsummpt2d 29781 cnre2csqima 29957 br2ndeq 31671 br2ndeqg 31673 filnetlem4 32376 |
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