Theorem dfmpt3 6014

Description: Alternate definition for the "maps to" notation df-mpt 4730. (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 4730	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
2		velsn 4193	. . . . . . 7 |- ( y e. { B } <-> y = B )
3	2	anbi2i 730	. . . . . 6 |- ( ( x e. A /\ y e. { B } ) <-> ( x e. A /\ y = B ) )
4	3	anbi2i 730	. . . . 5 |- ( ( z = <. x , y >. /\ ( x e. A /\ y e. { B } ) ) <-> ( z = <. x , y >. /\ ( x e. A /\ y = B ) ) )
5	4	2exbii 1775	. . . 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 5259	. . . 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 4983	. . . 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 292	. . 3 |- ( z e. U_ x e. A ( { x } X. { B } ) <-> z e. { <. x , y >. | ( x e. A /\ y = B ) } )
9	8	eqriv 2619	. 2 |- U_ x e. A ( { x } X. { B } ) = { <. x , y >. | ( x e. A /\ y = B ) }
10	1, 9	eqtr4i 2647	1 |- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } )

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {csn 4177   <.cop 4183   U_ciun 4520   {copab 4712   |-> cmpt 4729   X. cxp 5112

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121

This theorem is referenced by:  dfmpt  6410  taylpfval  24119  indval2  30076