Theorem dfmpt2 5786

Description: Alternate definition for the "maps to" notation df-mpt2 5460 (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 5766	. 2 |- (x e. A , y e. B |-> C) = (w e. (A X. B) |-> [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C)
2		vex 2554	. . . . 5 |- w e. _V
3		1stexg 5736	. . . . 5 |- (w e. _V -> ( 1st ` w) e. _V)
4	2, 3	ax-mp 7	. . . 4 |- ( 1st ` w) e. _V
5		2ndexg 5737	. . . . . 6 |- (w e. _V -> ( 2nd ` w) e. _V)
6	2, 5	ax-mp 7	. . . . 5 |- ( 2nd ` w) e. _V
7		dfmpt2.1	. . . . 5 |- C e. _V
8	6, 7	csbexa 3877	. . . 4 |- [_( 2nd ` w) / y ]_ C e. _V
9	4, 8	csbexa 3877	. . 3 |- [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C e. _V
10	9	dfmpt 5283	. 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 2175	. . . . 5 |- F/_xw
12		nfcsb1v 2876	. . . . 5 |- F/_x [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C
13	11, 12	nfop 3556	. . . 4 |- F/_x <.w , [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C >.
14	13	nfsn 3421	. . 3 |- F/_x { <.w , [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C >. }
15		nfcv 2175	. . . . 5 |- F/_yw
16		nfcv 2175	. . . . . 6 |- F/_y( 1st ` w)
17		nfcsb1v 2876	. . . . . 6 |- F/_y [_( 2nd ` w) / y ]_ C
18	16, 17	nfcsb 2878	. . . . 5 |- F/_y [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C
19	15, 18	nfop 3556	. . . 4 |- F/_y <.w , [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C >.
20	19	nfsn 3421	. . 3 |- F/_y { <.w , [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C >. }
21		nfcv 2175	. . 3 |- F/_w { <. <.x , y >. , C >. }
22		id 19	. . . . 5 |- (w = <.x , y >. -> w = <.x , y >.)
23		csbopeq1a 5756	. . . . 5 |- (w = <.x , y >. -> [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C = C)
24	22, 23	opeq12d 3548	. . . 4 |- (w = <.x , y >. -> <.w , [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C >. = <. <.x , y >. , C >.)
25	24	sneqd 3380	. . 3 |- (w = <.x , y >. -> { <.w , [_( 1st ` w) / x ]_ [_( 2nd ` w) / y ]_ C >. } = { <. <.x , y >. , C >. })
26	14, 20, 21, 25	iunxpf 4427	. 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 2061	1 |- (x e. A , y e. B |-> C) = U_x e. A U_y e. B { <. <.x , y >. , C >. }

Syntax hints:   = wceq 1242   e. wcel 1390   _Vcvv 2551   [_csb 2846   {csn 3367   <.cop 3370   U_ciun 3648   |-> cmpt 3809   X. cxp 4286   ` cfv 4845   |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710

This theorem is referenced by: (None)