Theorem dfmpo 6191

Description: Alternate definition for the maps-to notation df-mpo 5847 (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 6166	. 2 |- ( x e. A , y e. B |-> C ) = ( w e. ( A X. B ) |-> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C )
2		vex 2729	. . . . 5 |- w e. _V
3		1stexg 6135	. . . . 5 |- ( w e. _V -> ( 1st ` w ) e. _V )
4	2, 3	ax-mp 5	. . . 4 |- ( 1st ` w ) e. _V
5		2ndexg 6136	. . . . . 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 4111	. . . 4 |- [_ ( 2nd ` w ) / y ]_ C e. _V
9	4, 8	csbexa 4111	. . 3 |- [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C e. _V
10	9	dfmpt 5662	. 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 2308	. . . . 5 |- F/_ x w
12		nfcsb1v 3078	. . . . 5 |- F/_ x [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
13	11, 12	nfop 3774	. . . 4 |- F/_ x <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
14	13	nfsn 3636	. . 3 |- F/_ x { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
15		nfcv 2308	. . . . 5 |- F/_ y w
16		nfcv 2308	. . . . . 6 |- F/_ y ( 1st ` w )
17		nfcsb1v 3078	. . . . . 6 |- F/_ y [_ ( 2nd ` w ) / y ]_ C
18	16, 17	nfcsb 3082	. . . . 5 |- F/_ y [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
19	15, 18	nfop 3774	. . . 4 |- F/_ y <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
20	19	nfsn 3636	. . 3 |- F/_ y { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
21		nfcv 2308	. . 3 |- F/_ w { <. <. x , y >. , C >. }
22		id 19	. . . . 5 |- ( w = <. x , y >. -> w = <. x , y >. )
23		csbopeq1a 6156	. . . . 5 |- ( w = <. x , y >. -> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C = C )
24	22, 23	opeq12d 3766	. . . 4 |- ( w = <. x , y >. -> <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. = <. <. x , y >. , C >. )
25	24	sneqd 3589	. . 3 |- ( w = <. x , y >. -> { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. } = { <. <. x , y >. , C >. } )
26	14, 20, 21, 25	iunxpf 4752	. 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 2190	1 |- ( x e. A , y e. B |-> C ) = U_ x e. A U_ y e. B { <. <. x , y >. , C >. }

Syntax hints:   = wceq 1343   e. wcel 2136   _Vcvv 2726   [_csb 3045   {csn 3576   <.cop 3579   U_ciun 3866   |-> cmpt 4043   X. cxp 4602   ` cfv 5188   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109

This theorem is referenced by: (None)