Theorem dfopif 4399

Description: Rewrite df-op 4184 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	|- <. A , B >. = if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) )

Proof of Theorem dfopif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 4184	. 2 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
2		df-3an 1039	. . 3 |- ( ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) <-> ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) )
3	2	abbii 2739	. 2 |- { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } = { x | ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) }
4		iftrue 4092	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) ) = { { A } , { A , B } } )
5		ibar 525	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( x e. { { A } , { A , B } } <-> ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) ) )
6	5	abbi2dv 2742	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { { A } , { A , B } } = { x | ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) } )
7	4, 6	eqtr2d 2657	. . 3 |- ( ( A e. _V /\ B e. _V ) -> { x | ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) } = if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) ) )
8		pm2.21 120	. . . . . . 7 |- ( -. ( A e. _V /\ B e. _V ) -> ( ( A e. _V /\ B e. _V ) -> x e. (/) ) )
9	8	adantrd 484	. . . . . 6 |- ( -. ( A e. _V /\ B e. _V ) -> ( ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) -> x e. (/) ) )
10	9	abssdv 3676	. . . . 5 |- ( -. ( A e. _V /\ B e. _V ) -> { x | ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) } C_ (/) )
11		ss0 3974	. . . . 5 |- ( { x | ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) } C_ (/) -> { x | ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) } = (/) )
12	10, 11	syl 17	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> { x | ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) } = (/) )
13		iffalse 4095	. . . 4 |- ( -. ( A e. _V /\ B e. _V ) -> if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) ) = (/) )
14	12, 13	eqtr4d 2659	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> { x | ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) } = if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) ) )
15	7, 14	pm2.61i 176	. 2 |- { x | ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) } = if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) )
16	1, 3, 15	3eqtri 2648	1 |- <. A , B >. = if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) )

Syntax hints:   -. wn 3   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-op 4184

This theorem is referenced by:  dfopg  4400  opeq1  4402  opeq2  4403  nfop  4418  csbopg  4420  opprc  4424  opex  4932