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Theorem df2nd2 6437
 Description: An alternate possible definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
df2nd2
Distinct variable group:   ,,

Proof of Theorem df2nd2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fo2nd 6370 . . . . . 6
2 fofn 5658 . . . . . 6
31, 2ax-mp 5 . . . . 5
4 dffn5 5775 . . . . 5
53, 4mpbi 201 . . . 4
6 mptv 4304 . . . 4
75, 6eqtri 2458 . . 3
87reseq1i 5145 . 2
9 resopab 5190 . 2
10 vex 2961 . . . . 5
11 vex 2961 . . . . 5
1210, 11op2ndd 6361 . . . 4
1312eqeq2d 2449 . . 3
1413dfoprab3 6406 . 2
158, 9, 143eqtrri 2463 1
 Colors of variables: wff set class Syntax hints:   wa 360   wceq 1653   wcel 1726  cvv 2958  cop 3819  copab 4268   cmpt 4269   cxp 4879   cres 4883   wfn 5452  wfo 5455  cfv 5457  coprab 6085  c2nd 6351 This theorem is referenced by:  df2ndres  24097 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-oprab 6088  df-1st 6352  df-2nd 6353
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