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

Theorem df1st2 8081
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© ∣ š‘§ = š‘„} = (1st ↾ (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 7991 . . . . . 6 1st :V–onto→V
2 fofn 6800 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 dffn5 6943 . . . . 5 (1st Fn V ↔ 1st = (š‘¤ ∈ V ↦ (1st ā€˜š‘¤)))
53, 4mpbi 229 . . . 4 1st = (š‘¤ ∈ V ↦ (1st ā€˜š‘¤))
6 mptv 5257 . . . 4 (š‘¤ ∈ V ↦ (1st ā€˜š‘¤)) = {āŸØš‘¤, š‘§āŸ© ∣ š‘§ = (1st ā€˜š‘¤)}
75, 6eqtri 2754 . . 3 1st = {āŸØš‘¤, š‘§āŸ© ∣ š‘§ = (1st ā€˜š‘¤)}
87reseq1i 5970 . 2 (1st ↾ (V Ɨ V)) = ({āŸØš‘¤, š‘§āŸ© ∣ š‘§ = (1st ā€˜š‘¤)} ↾ (V Ɨ V))
9 resopab 6027 . 2 ({āŸØš‘¤, š‘§āŸ© ∣ š‘§ = (1st ā€˜š‘¤)} ↾ (V Ɨ V)) = {āŸØš‘¤, š‘§āŸ© ∣ (š‘¤ ∈ (V Ɨ V) ∧ š‘§ = (1st ā€˜š‘¤))}
10 vex 3472 . . . . 5 š‘„ ∈ V
11 vex 3472 . . . . 5 š‘¦ ∈ V
1210, 11op1std 7981 . . . 4 (š‘¤ = āŸØš‘„, š‘¦āŸ© → (1st ā€˜š‘¤) = š‘„)
1312eqeq2d 2737 . . 3 (š‘¤ = āŸØš‘„, š‘¦āŸ© → (š‘§ = (1st ā€˜š‘¤) ↔ š‘§ = š‘„))
1413dfoprab3 8036 . 2 {āŸØš‘¤, š‘§āŸ© ∣ (š‘¤ ∈ (V Ɨ V) ∧ š‘§ = (1st ā€˜š‘¤))} = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© ∣ š‘§ = š‘„}
158, 9, 143eqtrri 2759 1 {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© ∣ š‘§ = š‘„} = (1st ↾ (V Ɨ V))
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  āŸØcop 4629  {copab 5203   ↦ cmpt 5224   Ɨ cxp 5667   ↾ cres 5671   Fn wfn 6531  ā€“onto→wfo 6534  ā€˜cfv 6536  {coprab 7405  1st c1st 7969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-oprab 7408  df-1st 7971  df-2nd 7972
This theorem is referenced by:  df1stres  32430
  Copyright terms: Public domain W3C validator