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Mirrors > Home > MPE Home > Th. List > fo2nd | Structured version Visualization version GIF version |
Description: The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
fo2nd | ā¢ 2nd :VāontoāV |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vsnex 5387 | . . . . 5 ā¢ {š„} ā V | |
2 | 1 | rnex 7850 | . . . 4 ā¢ ran {š„} ā V |
3 | 2 | uniex 7679 | . . 3 ā¢ āŖ ran {š„} ā V |
4 | df-2nd 7923 | . . 3 ā¢ 2nd = (š„ ā V ā¦ āŖ ran {š„}) | |
5 | 3, 4 | fnmpti 6645 | . 2 ā¢ 2nd Fn V |
6 | 4 | rnmpt 5911 | . . 3 ā¢ ran 2nd = {š¦ ā£ āš„ ā V š¦ = āŖ ran {š„}} |
7 | vex 3450 | . . . . 5 ā¢ š¦ ā V | |
8 | opex 5422 | . . . . . 6 ā¢ āØš¦, š¦ā© ā V | |
9 | 7, 7 | op2nda 6181 | . . . . . . 7 ā¢ āŖ ran {āØš¦, š¦ā©} = š¦ |
10 | 9 | eqcomi 2746 | . . . . . 6 ā¢ š¦ = āŖ ran {āØš¦, š¦ā©} |
11 | sneq 4597 | . . . . . . . . 9 ā¢ (š„ = āØš¦, š¦ā© ā {š„} = {āØš¦, š¦ā©}) | |
12 | 11 | rneqd 5894 | . . . . . . . 8 ā¢ (š„ = āØš¦, š¦ā© ā ran {š„} = ran {āØš¦, š¦ā©}) |
13 | 12 | unieqd 4880 | . . . . . . 7 ā¢ (š„ = āØš¦, š¦ā© ā āŖ ran {š„} = āŖ ran {āØš¦, š¦ā©}) |
14 | 13 | rspceeqv 3596 | . . . . . 6 ā¢ ((āØš¦, š¦ā© ā V ā§ š¦ = āŖ ran {āØš¦, š¦ā©}) ā āš„ ā V š¦ = āŖ ran {š„}) |
15 | 8, 10, 14 | mp2an 691 | . . . . 5 ā¢ āš„ ā V š¦ = āŖ ran {š„} |
16 | 7, 15 | 2th 264 | . . . 4 ā¢ (š¦ ā V ā āš„ ā V š¦ = āŖ ran {š„}) |
17 | 16 | abbi2i 2874 | . . 3 ā¢ V = {š¦ ā£ āš„ ā V š¦ = āŖ ran {š„}} |
18 | 6, 17 | eqtr4i 2768 | . 2 ā¢ ran 2nd = V |
19 | df-fo 6503 | . 2 ā¢ (2nd :VāontoāV ā (2nd Fn V ā§ ran 2nd = V)) | |
20 | 5, 18, 19 | mpbir2an 710 | 1 ā¢ 2nd :VāontoāV |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ā wcel 2107 {cab 2714 āwrex 3074 Vcvv 3446 {csn 4587 āØcop 4593 āŖ cuni 4866 ran crn 5635 Fn wfn 6492 āontoāwfo 6495 2nd c2nd 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-fo 6503 df-2nd 7923 |
This theorem is referenced by: br2ndeqg 7945 2ndcof 7953 df2nd2 8032 2ndconst 8034 opco2 8057 iunfo 10476 cdaf 17937 2ndf1 18084 2ndf2 18085 2ndfcl 18087 gsum2dlem2 19749 upxp 22977 uptx 22979 cnmpt2nd 23023 uniiccdif 24945 xppreima 31565 2ndimaxp 31566 2ndresdju 31568 xppreima2 31570 2ndpreima 31624 fsuppcurry1 31645 gsummpt2d 31894 gsumpart 31900 cnre2csqima 32495 filnetlem4 34856 |
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