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Theorem dfopif 4214
Description: Rewrite df-op 4036 using if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
Assertion
Ref Expression
dfopif

Proof of Theorem dfopif
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-op 4036 . 2
2 df-3an 975 . . 3
32abbii 2591 . 2
4 iftrue 3947 . . . 4
5 ibar 504 . . . . 5
65abbi2dv 2594 . . . 4
74, 6eqtr2d 2499 . . 3
8 pm2.21 108 . . . . . . 7
98adantrd 468 . . . . . 6
109abssdv 3573 . . . . 5
11 ss0 3816 . . . . 5
1210, 11syl 16 . . . 4
13 iffalse 3950 . . . 4
1412, 13eqtr4d 2501 . . 3
157, 14pm2.61i 164 . 2
161, 3, 153eqtri 2490 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  {cab 2442   cvv 3109  C_wss 3475   c0 3784  ifcif 3941  {csn 4029  {cpr 4031  <.cop 4035
This theorem is referenced by:  dfopg  4215  opeq1  4217  opeq2  4218  nfop  4233  opprc  4239  opex  4716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-op 4036
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