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.

Step	Hyp	Ref	Expression
1		df-op 4036	. 2
2		df-3an 975	. . 3
3	2	abbii 2591	. 2
4		iftrue 3947	. . . 4
5		ibar 504	. . . . 5
6	5	abbi2dv 2594	. . . 4
7	4, 6	eqtr2d 2499	. . 3
8		pm2.21 108	. . . . . . 7
9	8	adantrd 468	. . . . . 6
10	9	abssdv 3573	. . . . 5
11		ss0 3816	. . . . 5
12	10, 11	syl 16	. . . 4
13		iffalse 3950	. . . 4
14	12, 13	eqtr4d 2501	. . 3
15	7, 14	pm2.61i 164	. 2
16	1, 3, 15	3eqtri 2490	1

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