Theorem dfmpo 6449

Description: Alternate definition for the maps-to notation df-mpo 6080 (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 6424	. 2 |- ( x e. A , y e. B |-> C ) = ( w e. ( A X. B ) |-> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C )
2		vex 2824	. . . . 5 |- w e. _V
3		1stexg 6391	. . . . 5 |- ( w e. _V -> ( 1st ` w ) e. _V )
4	2, 3	ax-mp 5	. . . 4 |- ( 1st ` w ) e. _V
5		2ndexg 6392	. . . . . 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 4257	. . . 4 |- [_ ( 2nd ` w ) / y ]_ C e. _V
9	4, 8	csbexa 4257	. . 3 |- [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C e. _V
10	9	dfmpt 5877	. 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 2392	. . . . 5 |- F/_ x w
12		nfcsb1v 3180	. . . . 5 |- F/_ x [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
13	11, 12	nfop 3915	. . . 4 |- F/_ x <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
14	13	nfsn 3765	. . 3 |- F/_ x { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
15		nfcv 2392	. . . . 5 |- F/_ y w
16		nfcv 2392	. . . . . 6 |- F/_ y ( 1st ` w )
17		nfcsb1v 3180	. . . . . 6 |- F/_ y [_ ( 2nd ` w ) / y ]_ C
18	16, 17	nfcsb 3185	. . . . 5 |- F/_ y [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
19	15, 18	nfop 3915	. . . 4 |- F/_ y <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
20	19	nfsn 3765	. . 3 |- F/_ y { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
21		nfcv 2392	. . 3 |- F/_ w { <. <. x , y >. , C >. }
22		id 19	. . . . 5 |- ( w = <. x , y >. -> w = <. x , y >. )
23		csbopeq1a 6412	. . . . 5 |- ( w = <. x , y >. -> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C = C )
24	22, 23	opeq12d 3907	. . . 4 |- ( w = <. x , y >. -> <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. = <. <. x , y >. , C >. )
25	24	sneqd 3718	. . 3 |- ( w = <. x , y >. -> { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. } = { <. <. x , y >. , C >. } )
26	14, 20, 21, 25	iunxpf 4923	. 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 2263	1 |- ( x e. A , y e. B |-> C ) = U_ x e. A U_ y e. B { <. <. x , y >. , C >. }

Syntax hints:   = wceq 1402   e. wcel 2209   _Vcvv 2821   [_csb 3147   {csn 3705   <.cop 3708   U_ciun 4007   |-> cmpt 4187   X. cxp 4767   ` cfv 5372   e. cmpo 6077   1stc1st 6362   2ndc2nd 6363

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  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  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-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365

This theorem is referenced by: (None)