MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df1st2 Structured version   Visualization version   GIF version

Theorem df1st2 8031
Description: An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
df1st2 {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ š‘§ = š‘„} = (1st ā†¾ (V Ɨ V))
Distinct variable group:   š‘„,š‘¦,š‘§

Proof of Theorem df1st2
Dummy variable š‘¤ is distinct from all other variables.
StepHypRef Expression
1 fo1st 7942 . . . . . 6 1st :Vā€“ontoā†’V
2 fofn 6759 . . . . . 6 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 dffn5 6902 . . . . 5 (1st Fn V ā†” 1st = (š‘¤ āˆˆ V ā†¦ (1st ā€˜š‘¤)))
53, 4mpbi 229 . . . 4 1st = (š‘¤ āˆˆ V ā†¦ (1st ā€˜š‘¤))
6 mptv 5222 . . . 4 (š‘¤ āˆˆ V ā†¦ (1st ā€˜š‘¤)) = {āŸØš‘¤, š‘§āŸ© āˆ£ š‘§ = (1st ā€˜š‘¤)}
75, 6eqtri 2761 . . 3 1st = {āŸØš‘¤, š‘§āŸ© āˆ£ š‘§ = (1st ā€˜š‘¤)}
87reseq1i 5934 . 2 (1st ā†¾ (V Ɨ V)) = ({āŸØš‘¤, š‘§āŸ© āˆ£ š‘§ = (1st ā€˜š‘¤)} ā†¾ (V Ɨ V))
9 resopab 5989 . 2 ({āŸØš‘¤, š‘§āŸ© āˆ£ š‘§ = (1st ā€˜š‘¤)} ā†¾ (V Ɨ V)) = {āŸØš‘¤, š‘§āŸ© āˆ£ (š‘¤ āˆˆ (V Ɨ V) āˆ§ š‘§ = (1st ā€˜š‘¤))}
10 vex 3448 . . . . 5 š‘„ āˆˆ V
11 vex 3448 . . . . 5 š‘¦ āˆˆ V
1210, 11op1std 7932 . . . 4 (š‘¤ = āŸØš‘„, š‘¦āŸ© ā†’ (1st ā€˜š‘¤) = š‘„)
1312eqeq2d 2744 . . 3 (š‘¤ = āŸØš‘„, š‘¦āŸ© ā†’ (š‘§ = (1st ā€˜š‘¤) ā†” š‘§ = š‘„))
1413dfoprab3 7987 . 2 {āŸØš‘¤, š‘§āŸ© āˆ£ (š‘¤ āˆˆ (V Ɨ V) āˆ§ š‘§ = (1st ā€˜š‘¤))} = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ š‘§ = š‘„}
158, 9, 143eqtrri 2766 1 {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ š‘§ = š‘„} = (1st ā†¾ (V Ɨ V))
Colors of variables: wff setvar class
Syntax hints:   āˆ§ wa 397   = wceq 1542   āˆˆ wcel 2107  Vcvv 3444  āŸØcop 4593  {copab 5168   ā†¦ cmpt 5189   Ɨ cxp 5632   ā†¾ cres 5636   Fn wfn 6492  ā€“ontoā†’wfo 6495  ā€˜cfv 6497  {coprab 7359  1st c1st 7920
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 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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  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-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-oprab 7362  df-1st 7922  df-2nd 7923
This theorem is referenced by:  df1stres  31664
  Copyright terms: Public domain W3C validator