Theorem dfmpo 6397

Description: Alternate definition for the maps-to notation df-mpo 6033 (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 6372	. 2 |- ( x e. A , y e. B |-> C ) = ( w e. ( A X. B ) |-> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C )
2		vex 2806	. . . . 5 |- w e. _V
3		1stexg 6339	. . . . 5 |- ( w e. _V -> ( 1st ` w ) e. _V )
4	2, 3	ax-mp 5	. . . 4 |- ( 1st ` w ) e. _V
5		2ndexg 6340	. . . . . 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 4223	. . . 4 |- [_ ( 2nd ` w ) / y ]_ C e. _V
9	4, 8	csbexa 4223	. . 3 |- [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C e. _V
10	9	dfmpt 5833	. 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 2375	. . . . 5 |- F/_ x w
12		nfcsb1v 3161	. . . . 5 |- F/_ x [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
13	11, 12	nfop 3883	. . . 4 |- F/_ x <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
14	13	nfsn 3733	. . 3 |- F/_ x { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
15		nfcv 2375	. . . . 5 |- F/_ y w
16		nfcv 2375	. . . . . 6 |- F/_ y ( 1st ` w )
17		nfcsb1v 3161	. . . . . 6 |- F/_ y [_ ( 2nd ` w ) / y ]_ C
18	16, 17	nfcsb 3166	. . . . 5 |- F/_ y [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
19	15, 18	nfop 3883	. . . 4 |- F/_ y <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
20	19	nfsn 3733	. . 3 |- F/_ y { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
21		nfcv 2375	. . 3 |- F/_ w { <. <. x , y >. , C >. }
22		id 19	. . . . 5 |- ( w = <. x , y >. -> w = <. x , y >. )
23		csbopeq1a 6360	. . . . 5 |- ( w = <. x , y >. -> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C = C )
24	22, 23	opeq12d 3875	. . . 4 |- ( w = <. x , y >. -> <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. = <. <. x , y >. , C >. )
25	24	sneqd 3686	. . 3 |- ( w = <. x , y >. -> { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. } = { <. <. x , y >. , C >. } )
26	14, 20, 21, 25	iunxpf 4884	. 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 2256	1 |- ( x e. A , y e. B |-> C ) = U_ x e. A U_ y e. B { <. <. x , y >. , C >. }

Syntax hints:   = wceq 1398   e. wcel 2202   _Vcvv 2803   [_csb 3128   {csn 3673   <.cop 3676   U_ciun 3975   |-> cmpt 4155   X. cxp 4729   ` cfv 5333   e. cmpo 6030   1stc1st 6310   2ndc2nd 6311

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  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-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313

This theorem is referenced by: (None)