Theorem dfmpt3 5462

Description: Alternate definition for the maps-to notation df-mpt 4157. (Contributed by Mario Carneiro, 30-Dec-2016.)

Assertion

Ref	Expression
dfmpt3	|- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } )

Proof of Theorem dfmpt3

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 4157	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
2		velsn 3690	. . . . . . 7 |- ( y e. { B } <-> y = B )
3	2	anbi2i 457	. . . . . 6 |- ( ( x e. A /\ y e. { B } ) <-> ( x e. A /\ y = B ) )
4	3	anbi2i 457	. . . . 5 |- ( ( z = <. x , y >. /\ ( x e. A /\ y e. { B } ) ) <-> ( z = <. x , y >. /\ ( x e. A /\ y = B ) ) )
5	4	2exbii 1655	. . . 4 |- ( E. x E. y ( z = <. x , y >. /\ ( x e. A /\ y e. { B } ) ) <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ y = B ) ) )
6		eliunxp 4875	. . . 4 |- ( z e. U_ x e. A ( { x } X. { B } ) <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ y e. { B } ) ) )
7		elopab 4358	. . . 4 |- ( z e. { <. x , y >. | ( x e. A /\ y = B ) } <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ y = B ) ) )
8	5, 6, 7	3bitr4i 212	. . 3 |- ( z e. U_ x e. A ( { x } X. { B } ) <-> z e. { <. x , y >. | ( x e. A /\ y = B ) } )
9	8	eqriv 2228	. 2 |- U_ x e. A ( { x } X. { B } ) = { <. x , y >. | ( x e. A /\ y = B ) }
10	1, 9	eqtr4i 2255	1 |- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } )

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2202   {csn 3673   <.cop 3676   U_ciun 3975   {copab 4154   |-> cmpt 4155   X. cxp 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-iun 3977  df-opab 4156  df-mpt 4157  df-xp 4737  df-rel 4738

This theorem is referenced by:  dfmpt  5833  dfmptg  5835