Theorem dfmpt2 7267

Description: Alternate definition for the "maps to" notation df-mpt2 6655 (although it requires that C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfmpt2.1	|- C e. _V

Assertion

Ref	Expression
dfmpt2	|- ( 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 dfmpt2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpt2mpts 7234	. 2 |- ( x e. A , y e. B |-> C ) = ( w e. ( A X. B ) |-> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C )
2		dfmpt2.1	. . . . 5 |- C e. _V
3	2	csbex 4793	. . . 4 |- [_ ( 2nd ` w ) / y ]_ C e. _V
4	3	csbex 4793	. . 3 |- [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C e. _V
5	4	dfmpt 6410	. 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 >. }
6		nfcv 2764	. . . . 5 |- F/_ x w
7		nfcsb1v 3549	. . . . 5 |- F/_ x [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
8	6, 7	nfop 4418	. . . 4 |- F/_ x <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
9	8	nfsn 4242	. . 3 |- F/_ x { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
10		nfcv 2764	. . . . 5 |- F/_ y w
11		nfcv 2764	. . . . . 6 |- F/_ y ( 1st ` w )
12		nfcsb1v 3549	. . . . . 6 |- F/_ y [_ ( 2nd ` w ) / y ]_ C
13	11, 12	nfcsb 3551	. . . . 5 |- F/_ y [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C
14	10, 13	nfop 4418	. . . 4 |- F/_ y <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >.
15	14	nfsn 4242	. . 3 |- F/_ y { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. }
16		nfcv 2764	. . 3 |- F/_ w { <. <. x , y >. , C >. }
17		id 22	. . . . 5 |- ( w = <. x , y >. -> w = <. x , y >. )
18		csbopeq1a 7221	. . . . 5 |- ( w = <. x , y >. -> [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C = C )
19	17, 18	opeq12d 4410	. . . 4 |- ( w = <. x , y >. -> <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. = <. <. x , y >. , C >. )
20	19	sneqd 4189	. . 3 |- ( w = <. x , y >. -> { <. w , [_ ( 1st ` w ) / x ]_ [_ ( 2nd ` w ) / y ]_ C >. } = { <. <. x , y >. , C >. } )
21	9, 15, 16, 20	iunxpf 5270	. 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 >. }
22	1, 5, 21	3eqtri 2648	1 |- ( x e. A , y e. B |-> C ) = U_ x e. A U_ y e. B { <. <. x , y >. , C >. }

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   {csn 4177   <.cop 4183   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   ` cfv 5888   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  fpar  7281