Theorem dfmpt3 5501

Description: Alternate definition for the maps-to notation df-mpt 4189. (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 4189	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
2		velsn 3722	. . . . . . 7 |- ( y e. { B } <-> y = B )
3	2	anbi2i 461	. . . . . 6 |- ( ( x e. A /\ y e. { B } ) <-> ( x e. A /\ y = B ) )
4	3	anbi2i 461	. . . . 5 |- ( ( z = <. x , y >. /\ ( x e. A /\ y e. { B } ) ) <-> ( z = <. x , y >. /\ ( x e. A /\ y = B ) ) )
5	4	2exbii 1659	. . . 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 4914	. . . 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 4395	. . . 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 2235	. 2 |- U_ x e. A ( { x } X. { B } ) = { <. x , y >. | ( x e. A /\ y = B ) }
10	1, 9	eqtr4i 2262	1 |- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } )

Syntax hints:   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   {csn 3705   <.cop 3708   U_ciun 4007   {copab 4186   |-> cmpt 4187   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-iun 4009  df-opab 4188  df-mpt 4189  df-xp 4775  df-rel 4776

This theorem is referenced by:  dfmpt  5877  dfmptg  5879