![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fo1st | Structured version Visualization version GIF version |
Description: The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
fo1st | ā¢ 1st :VāontoāV |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vsnex 5430 | . . . . 5 ā¢ {š„} ā V | |
2 | 1 | dmex 7902 | . . . 4 ā¢ dom {š„} ā V |
3 | 2 | uniex 7731 | . . 3 ā¢ āŖ dom {š„} ā V |
4 | df-1st 7975 | . . 3 ā¢ 1st = (š„ ā V ā¦ āŖ dom {š„}) | |
5 | 3, 4 | fnmpti 6694 | . 2 ā¢ 1st Fn V |
6 | 4 | rnmpt 5955 | . . 3 ā¢ ran 1st = {š¦ ā£ āš„ ā V š¦ = āŖ dom {š„}} |
7 | vex 3479 | . . . . 5 ā¢ š¦ ā V | |
8 | opex 5465 | . . . . . 6 ā¢ āØš¦, š¦ā© ā V | |
9 | 7, 7 | op1sta 6225 | . . . . . . 7 ā¢ āŖ dom {āØš¦, š¦ā©} = š¦ |
10 | 9 | eqcomi 2742 | . . . . . 6 ā¢ š¦ = āŖ dom {āØš¦, š¦ā©} |
11 | sneq 4639 | . . . . . . . . 9 ā¢ (š„ = āØš¦, š¦ā© ā {š„} = {āØš¦, š¦ā©}) | |
12 | 11 | dmeqd 5906 | . . . . . . . 8 ā¢ (š„ = āØš¦, š¦ā© ā dom {š„} = dom {āØš¦, š¦ā©}) |
13 | 12 | unieqd 4923 | . . . . . . 7 ā¢ (š„ = āØš¦, š¦ā© ā āŖ dom {š„} = āŖ dom {āØš¦, š¦ā©}) |
14 | 13 | rspceeqv 3634 | . . . . . 6 ā¢ ((āØš¦, š¦ā© ā V ā§ š¦ = āŖ dom {āØš¦, š¦ā©}) ā āš„ ā V š¦ = āŖ dom {š„}) |
15 | 8, 10, 14 | mp2an 691 | . . . . 5 ā¢ āš„ ā V š¦ = āŖ dom {š„} |
16 | 7, 15 | 2th 264 | . . . 4 ā¢ (š¦ ā V ā āš„ ā V š¦ = āŖ dom {š„}) |
17 | 16 | eqabi 2870 | . . 3 ā¢ V = {š¦ ā£ āš„ ā V š¦ = āŖ dom {š„}} |
18 | 6, 17 | eqtr4i 2764 | . 2 ā¢ ran 1st = V |
19 | df-fo 6550 | . 2 ā¢ (1st :VāontoāV ā (1st Fn V ā§ ran 1st = V)) | |
20 | 5, 18, 19 | mpbir2an 710 | 1 ā¢ 1st :VāontoāV |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ā wcel 2107 {cab 2710 āwrex 3071 Vcvv 3475 {csn 4629 āØcop 4635 āŖ cuni 4909 dom cdm 5677 ran crn 5678 Fn wfn 6539 āontoāwfo 6542 1st c1st 7973 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-fun 6546 df-fn 6547 df-fo 6550 df-1st 7975 |
This theorem is referenced by: br1steqg 7997 1stcof 8005 df1st2 8084 1stconst 8086 fsplit 8103 opco1 8109 fpwwe 10641 axpre-sup 11164 homadm 17990 homacd 17991 dmaf 17999 cdaf 18000 1stf1 18144 1stf2 18145 1stfcl 18149 upxp 23127 uptx 23129 cnmpt1st 23172 bcthlem4 24844 uniiccdif 25095 precsexlem10 27662 precsexlem11 27663 vafval 29856 smfval 29858 0vfval 29859 vsfval 29886 xppreima 31871 xppreima2 31876 1stpreimas 31927 1stpreima 31928 fsuppcurry2 31951 gsummpt2d 32201 cnre2csqima 32891 poimirlem26 36514 poimirlem27 36515 |
Copyright terms: Public domain | W3C validator |