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Theorem fo1st 7995
Description: The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fo1st 1st :Vā€“ontoā†’V

Proof of Theorem fo1st
Dummy variables š‘„ š‘¦ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vsnex 5430 . . . . 5 {š‘„} āˆˆ V
21dmex 7902 . . . 4 dom {š‘„} āˆˆ V
32uniex 7731 . . 3 āˆŖ dom {š‘„} āˆˆ V
4 df-1st 7975 . . 3 1st = (š‘„ āˆˆ V ā†¦ āˆŖ dom {š‘„})
53, 4fnmpti 6694 . 2 1st Fn V
64rnmpt 5955 . . 3 ran 1st = {š‘¦ āˆ£ āˆƒš‘„ āˆˆ V š‘¦ = āˆŖ dom {š‘„}}
7 vex 3479 . . . . 5 š‘¦ āˆˆ V
8 opex 5465 . . . . . 6 āŸØš‘¦, š‘¦āŸ© āˆˆ V
97, 7op1sta 6225 . . . . . . 7 āˆŖ dom {āŸØš‘¦, š‘¦āŸ©} = š‘¦
109eqcomi 2742 . . . . . 6 š‘¦ = āˆŖ dom {āŸØš‘¦, š‘¦āŸ©}
11 sneq 4639 . . . . . . . . 9 (š‘„ = āŸØš‘¦, š‘¦āŸ© ā†’ {š‘„} = {āŸØš‘¦, š‘¦āŸ©})
1211dmeqd 5906 . . . . . . . 8 (š‘„ = āŸØš‘¦, š‘¦āŸ© ā†’ dom {š‘„} = dom {āŸØš‘¦, š‘¦āŸ©})
1312unieqd 4923 . . . . . . 7 (š‘„ = āŸØš‘¦, š‘¦āŸ© ā†’ āˆŖ dom {š‘„} = āˆŖ dom {āŸØš‘¦, š‘¦āŸ©})
1413rspceeqv 3634 . . . . . 6 ((āŸØš‘¦, š‘¦āŸ© āˆˆ V āˆ§ š‘¦ = āˆŖ dom {āŸØš‘¦, š‘¦āŸ©}) ā†’ āˆƒš‘„ āˆˆ V š‘¦ = āˆŖ dom {š‘„})
158, 10, 14mp2an 691 . . . . 5 āˆƒš‘„ āˆˆ V š‘¦ = āˆŖ dom {š‘„}
167, 152th 264 . . . 4 (š‘¦ āˆˆ V ā†” āˆƒš‘„ āˆˆ V š‘¦ = āˆŖ dom {š‘„})
1716eqabi 2870 . . 3 V = {š‘¦ āˆ£ āˆƒš‘„ āˆˆ V š‘¦ = āˆŖ dom {š‘„}}
186, 17eqtr4i 2764 . 2 ran 1st = V
19 df-fo 6550 . 2 (1st :Vā€“ontoā†’V ā†” (1st Fn V āˆ§ ran 1st = V))
205, 18, 19mpbir2an 710 1 1st :Vā€“ontoā†’V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   āˆˆ wcel 2107  {cab 2710  āˆƒwrex 3071  Vcvv 3475  {csn 4629  āŸØcop 4635  āˆŖ cuni 4909  dom cdm 5677  ran crn 5678   Fn wfn 6539  ā€“ontoā†’wfo 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
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