Theorem fsplit 7282

Description: A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 7281 in order to build compound functions such as y = ( ( sqr ` x ) + ( abs ` x ) ). (Contributed by NM, 17-Sep-2007.)

Assertion

Ref	Expression
fsplit	|- `' ( 1st |` _I ) = ( x e. _V |-> <. x , x >. )

Proof of Theorem fsplit

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- x e. _V
2		vex 3203	. . . . 5 |- y e. _V
3	1, 2	brcnv 5305	. . . 4 |- ( x `' ( 1st |` _I ) y <-> y ( 1st |` _I ) x )
4	1	brres 5402	. . . . 5 |- ( y ( 1st |` _I ) x <-> ( y 1st x /\ y e. _I ) )
5		19.42v 1918	. . . . . . 7 |- ( E. z ( ( 1st ` y ) = x /\ y = <. z , z >. ) <-> ( ( 1st ` y ) = x /\ E. z y = <. z , z >. ) )
6		vex 3203	. . . . . . . . . . 11 |- z e. _V
7	6, 6	op1std 7178	. . . . . . . . . 10 |- ( y = <. z , z >. -> ( 1st ` y ) = z )
8	7	eqeq1d 2624	. . . . . . . . 9 |- ( y = <. z , z >. -> ( ( 1st ` y ) = x <-> z = x ) )
9	8	pm5.32ri 670	. . . . . . . 8 |- ( ( ( 1st ` y ) = x /\ y = <. z , z >. ) <-> ( z = x /\ y = <. z , z >. ) )
10	9	exbii 1774	. . . . . . 7 |- ( E. z ( ( 1st ` y ) = x /\ y = <. z , z >. ) <-> E. z ( z = x /\ y = <. z , z >. ) )
11		fo1st 7188	. . . . . . . . . 10 |- 1st : _V -onto-> _V
12		fofn 6117	. . . . . . . . . 10 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
13	11, 12	ax-mp 5	. . . . . . . . 9 |- 1st Fn _V
14		fnbrfvb 6236	. . . . . . . . 9 |- ( ( 1st Fn _V /\ y e. _V ) -> ( ( 1st ` y ) = x <-> y 1st x ) )
15	13, 2, 14	mp2an 708	. . . . . . . 8 |- ( ( 1st ` y ) = x <-> y 1st x )
16		dfid2 5027	. . . . . . . . . 10 |- _I = { <. z , z >. | z = z }
17	16	eleq2i 2693	. . . . . . . . 9 |- ( y e. _I <-> y e. { <. z , z >. | z = z } )
18		nfe1 2027	. . . . . . . . . . 11 |- F/ z E. z ( y = <. z , z >. /\ z = z )
19	18	19.9 2072	. . . . . . . . . 10 |- ( E. z E. z ( y = <. z , z >. /\ z = z ) <-> E. z ( y = <. z , z >. /\ z = z ) )
20		elopab 4983	. . . . . . . . . 10 |- ( y e. { <. z , z >. | z = z } <-> E. z E. z ( y = <. z , z >. /\ z = z ) )
21		equid 1939	. . . . . . . . . . . 12 |- z = z
22	21	biantru 526	. . . . . . . . . . 11 |- ( y = <. z , z >. <-> ( y = <. z , z >. /\ z = z ) )
23	22	exbii 1774	. . . . . . . . . 10 |- ( E. z y = <. z , z >. <-> E. z ( y = <. z , z >. /\ z = z ) )
24	19, 20, 23	3bitr4i 292	. . . . . . . . 9 |- ( y e. { <. z , z >. | z = z } <-> E. z y = <. z , z >. )
25	17, 24	bitr2i 265	. . . . . . . 8 |- ( E. z y = <. z , z >. <-> y e. _I )
26	15, 25	anbi12i 733	. . . . . . 7 |- ( ( ( 1st ` y ) = x /\ E. z y = <. z , z >. ) <-> ( y 1st x /\ y e. _I ) )
27	5, 10, 26	3bitr3ri 291	. . . . . 6 |- ( ( y 1st x /\ y e. _I ) <-> E. z ( z = x /\ y = <. z , z >. ) )
28		id 22	. . . . . . . . 9 |- ( z = x -> z = x )
29	28, 28	opeq12d 4410	. . . . . . . 8 |- ( z = x -> <. z , z >. = <. x , x >. )
30	29	eqeq2d 2632	. . . . . . 7 |- ( z = x -> ( y = <. z , z >. <-> y = <. x , x >. ) )
31	1, 30	ceqsexv 3242	. . . . . 6 |- ( E. z ( z = x /\ y = <. z , z >. ) <-> y = <. x , x >. )
32	27, 31	bitri 264	. . . . 5 |- ( ( y 1st x /\ y e. _I ) <-> y = <. x , x >. )
33	4, 32	bitri 264	. . . 4 |- ( y ( 1st |` _I ) x <-> y = <. x , x >. )
34	3, 33	bitri 264	. . 3 |- ( x `' ( 1st |` _I ) y <-> y = <. x , x >. )
35	34	opabbii 4717	. 2 |- { <. x , y >. | x `' ( 1st |` _I ) y } = { <. x , y >. | y = <. x , x >. }
36		relcnv 5503	. . 3 |- Rel `' ( 1st |` _I )
37		dfrel4v 5584	. . 3 |- ( Rel `' ( 1st |` _I ) <-> `' ( 1st |` _I ) = { <. x , y >. | x `' ( 1st |` _I ) y } )
38	36, 37	mpbi 220	. 2 |- `' ( 1st |` _I ) = { <. x , y >. | x `' ( 1st |` _I ) y }
39		mptv 4751	. 2 |- ( x e. _V |-> <. x , x >. ) = { <. x , y >. | y = <. x , x >. }
40	35, 38, 39	3eqtr4i 2654	1 |- `' ( 1st |` _I ) = ( x e. _V |-> <. x , x >. )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   {copab 4712   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   Rel wrel 5119   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888   1stc1st 7166

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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168

This theorem is referenced by: (None)