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Theorem df1st2 8107
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 8017 . . . . . 6 1st :V–onto→V
2 fofn 6816 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 dffn5 6960 . . . . 5 (1st Fn V ↔ 1st = (š‘¤ ∈ V ↦ (1st ā€˜š‘¤)))
53, 4mpbi 229 . . . 4 1st = (š‘¤ ∈ V ↦ (1st ā€˜š‘¤))
6 mptv 5266 . . . 4 (š‘¤ ∈ V ↦ (1st ā€˜š‘¤)) = {āŸØš‘¤, š‘§āŸ© ∣ š‘§ = (1st ā€˜š‘¤)}
75, 6eqtri 2755 . . 3 1st = {āŸØš‘¤, š‘§āŸ© ∣ š‘§ = (1st ā€˜š‘¤)}
87reseq1i 5983 . 2 (1st ↾ (V Ɨ V)) = ({āŸØš‘¤, š‘§āŸ© ∣ š‘§ = (1st ā€˜š‘¤)} ↾ (V Ɨ V))
9 resopab 6041 . 2 ({āŸØš‘¤, š‘§āŸ© ∣ š‘§ = (1st ā€˜š‘¤)} ↾ (V Ɨ V)) = {āŸØš‘¤, š‘§āŸ© ∣ (š‘¤ ∈ (V Ɨ V) ∧ š‘§ = (1st ā€˜š‘¤))}
10 vex 3475 . . . . 5 š‘„ ∈ V
11 vex 3475 . . . . 5 š‘¦ ∈ V
1210, 11op1std 8007 . . . 4 (š‘¤ = āŸØš‘„, š‘¦āŸ© → (1st ā€˜š‘¤) = š‘„)
1312eqeq2d 2738 . . 3 (š‘¤ = āŸØš‘„, š‘¦āŸ© → (š‘§ = (1st ā€˜š‘¤) ↔ š‘§ = š‘„))
1413dfoprab3 8062 . 2 {āŸØš‘¤, š‘§āŸ© ∣ (š‘¤ ∈ (V Ɨ V) ∧ š‘§ = (1st ā€˜š‘¤))} = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© ∣ š‘§ = š‘„}
158, 9, 143eqtrri 2760 1 {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© ∣ š‘§ = š‘„} = (1st ↾ (V Ɨ V))
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3471  āŸØcop 4636  {copab 5212   ↦ cmpt 5233   Ɨ cxp 5678   ↾ cres 5682   Fn wfn 6546  ā€“onto→wfo 6549  ā€˜cfv 6551  {coprab 7425  1st c1st 7995
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fo 6557  df-fv 6559  df-oprab 7428  df-1st 7997  df-2nd 7998
This theorem is referenced by:  df1stres  32501
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