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Theorem df1st2 6425
 Description: An alternate possible definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
df1st2
Distinct variable group:   ,,

Proof of Theorem df1st2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fo1st 6358 . . . . . 6
2 fofn 5647 . . . . . 6
31, 2ax-mp 8 . . . . 5
4 dffn5 5764 . . . . 5
53, 4mpbi 200 . . . 4
6 mptv 4293 . . . 4
75, 6eqtri 2455 . . 3
87reseq1i 5134 . 2
9 resopab 5179 . 2
10 vex 2951 . . . . 5
11 vex 2951 . . . . 5
1210, 11op1std 6349 . . . 4
1312eqeq2d 2446 . . 3
1413dfoprab3 6395 . 2
158, 9, 143eqtrri 2460 1
 Colors of variables: wff set class Syntax hints:   wa 359   wceq 1652   wcel 1725  cvv 2948  cop 3809  copab 4257   cmpt 4258   cxp 4868   cres 4872   wfn 5441  wfo 5444  cfv 5446  coprab 6074  c1st 6339 This theorem is referenced by:  df1stres  24083 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-oprab 6077  df-1st 6341  df-2nd 6342
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