Theorem dfmpo 6171

Description: Alternate definition for the maps-to notation df-mpo 5830 (although it requires that C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfmpo.1	|- C e. _V

Assertion

Ref	Expression
dfmpo	|- ( x e. A , y e. B |-> C ) = U_ x e. A U_ y e. B { <. <. x , y >. , C >. }

Distinct variable groups:   x, y, A   x, B, y

Allowed substitution hints:   C( x, y)

Proof of Theorem dfmpo

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpompts 6147	. 2 |- ( x e. A , y e. B |-> C ) = ( w e. ( A X. B ) |-> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C )
2		vex 2715	. . . . 5 |- w e. _V
3		1stexg 6116	. . . . 5 |- ( w e. _V -> ( 1st ` w ) e. _V )
4	2, 3	ax-mp 5	. . . 4 |- ( 1st ` w ) e. _V
5		2ndexg 6117	. . . . . 6 |- ( w e. _V -> ( 2nd ` w ) e. _V )
6	2, 5	ax-mp 5	. . . . 5 |- ( 2nd ` w ) e. _V
7		dfmpo.1	. . . . 5 |- C e. _V
8	6, 7	csbexa 4094	. . . 4 |- [_ ( 2nd ` w ) / y ]_ C e. _V
9	4, 8	csbexa 4094	. . 3 |- [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C e. _V
10	9	dfmpt 5645	. 2 |- ( w e. ( A X. B ) |-> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C ) = U_ w e. ( A X. B ) { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
11		nfcv 2299	. . . . 5 |- F/_ x w
12		nfcsb1v 3064	. . . . 5 |- F/_ x [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
13	11, 12	nfop 3758	. . . 4 |- F/_ x <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
14	13	nfsn 3620	. . 3 |- F/_ x { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
15		nfcv 2299	. . . . 5 |- F/_ y w
16		nfcv 2299	. . . . . 6 |- F/_ y ( 1st ` w )
17		nfcsb1v 3064	. . . . . 6 |- F/_ y [_ ( 2nd ` w ) / y ]_ C
18	16, 17	nfcsb 3068	. . . . 5 |- F/_ y [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
19	15, 18	nfop 3758	. . . 4 |- F/_ y <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
20	19	nfsn 3620	. . 3 |- F/_ y { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
21		nfcv 2299	. . 3 |- F/_ w { <. <. x , y >. , C >. }
22		id 19	. . . . 5 |- ( w = <. x , y >. -> w = <. x , y >. )
23		csbopeq1a 6137	. . . . 5 |- ( w = <. x , y >. -> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C = C )
24	22, 23	opeq12d 3750	. . . 4 |- ( w = <. x , y >. -> <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. = <. <. x , y >. , C >. )
25	24	sneqd 3573	. . 3 |- ( w = <. x , y >. -> { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. } = { <. <. x , y >. , C >. } )
26	14, 20, 21, 25	iunxpf 4735	. 2 |- U_ w e. ( A X. B ) { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. } = U_ x e. A U_ y e. B { <. <. x , y >. , C >. }
27	1, 10, 26	3eqtri 2182	1 |- ( x e. A , y e. B |-> C ) = U_ x e. A U_ y e. B { <. <. x , y >. , C >. }

Syntax hints:   = wceq 1335   e. wcel 2128   _Vcvv 2712   [_csb 3031   {csn 3560   <.cop 3563   U_ciun 3850   |-> cmpt 4026   X. cxp 4585   ` cfv 5171   e. cmpo 5827   1stc1st 6087   2ndc2nd 6088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090

This theorem is referenced by: (None)