Theorem dfrn5 31677

Description: Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
dfrn5	|- ran A = ( ( 2nd |` ( _V X. _V ) ) " A )

Proof of Theorem dfrn5

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

Step	Hyp	Ref	Expression
1		excom 2042	. . . 4 |- ( E. y E. p E. z ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) <-> E. p E. y E. z ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) )
2		opex 4932	. . . . . . . 8 |- <. y , z >. e. _V
3		breq1 4656	. . . . . . . . . 10 |- ( p = <. y , z >. -> ( p 2nd x <-> <. y , z >. 2nd x ) )
4		eleq1 2689	. . . . . . . . . 10 |- ( p = <. y , z >. -> ( p e. A <-> <. y , z >. e. A ) )
5	3, 4	anbi12d 747	. . . . . . . . 9 |- ( p = <. y , z >. -> ( ( p 2nd x /\ p e. A ) <-> ( <. y , z >. 2nd x /\ <. y , z >. e. A ) ) )
6		vex 3203	. . . . . . . . . . . 12 |- y e. _V
7		vex 3203	. . . . . . . . . . . 12 |- z e. _V
8	6, 7	br2ndeq 31671	. . . . . . . . . . 11 |- ( <. y , z >. 2nd x <-> x = z )
9		equcom 1945	. . . . . . . . . . 11 |- ( x = z <-> z = x )
10	8, 9	bitri 264	. . . . . . . . . 10 |- ( <. y , z >. 2nd x <-> z = x )
11	10	anbi1i 731	. . . . . . . . 9 |- ( ( <. y , z >. 2nd x /\ <. y , z >. e. A ) <-> ( z = x /\ <. y , z >. e. A ) )
12	5, 11	syl6bb 276	. . . . . . . 8 |- ( p = <. y , z >. -> ( ( p 2nd x /\ p e. A ) <-> ( z = x /\ <. y , z >. e. A ) ) )
13	2, 12	ceqsexv 3242	. . . . . . 7 |- ( E. p ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) <-> ( z = x /\ <. y , z >. e. A ) )
14	13	exbii 1774	. . . . . 6 |- ( E. z E. p ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) <-> E. z ( z = x /\ <. y , z >. e. A ) )
15		excom 2042	. . . . . 6 |- ( E. z E. p ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) <-> E. p E. z ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) )
16		vex 3203	. . . . . . 7 |- x e. _V
17		opeq2 4403	. . . . . . . 8 |- ( z = x -> <. y , z >. = <. y , x >. )
18	17	eleq1d 2686	. . . . . . 7 |- ( z = x -> ( <. y , z >. e. A <-> <. y , x >. e. A ) )
19	16, 18	ceqsexv 3242	. . . . . 6 |- ( E. z ( z = x /\ <. y , z >. e. A ) <-> <. y , x >. e. A )
20	14, 15, 19	3bitr3ri 291	. . . . 5 |- ( <. y , x >. e. A <-> E. p E. z ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) )
21	20	exbii 1774	. . . 4 |- ( E. y <. y , x >. e. A <-> E. y E. p E. z ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) )
22		ancom 466	. . . . . 6 |- ( ( p e. A /\ p ( 2nd |` ( _V X. _V ) ) x ) <-> ( p ( 2nd |` ( _V X. _V ) ) x /\ p e. A ) )
23		anass 681	. . . . . . 7 |- ( ( ( E. y E. z p = <. y , z >. /\ p 2nd x ) /\ p e. A ) <-> ( E. y E. z p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) )
24	16	brres 5402	. . . . . . . . 9 |- ( p ( 2nd |` ( _V X. _V ) ) x <-> ( p 2nd x /\ p e. ( _V X. _V ) ) )
25		ancom 466	. . . . . . . . . 10 |- ( ( p 2nd x /\ p e. ( _V X. _V ) ) <-> ( p e. ( _V X. _V ) /\ p 2nd x ) )
26		elvv 5177	. . . . . . . . . . 11 |- ( p e. ( _V X. _V ) <-> E. y E. z p = <. y , z >. )
27	26	anbi1i 731	. . . . . . . . . 10 |- ( ( p e. ( _V X. _V ) /\ p 2nd x ) <-> ( E. y E. z p = <. y , z >. /\ p 2nd x ) )
28	25, 27	bitri 264	. . . . . . . . 9 |- ( ( p 2nd x /\ p e. ( _V X. _V ) ) <-> ( E. y E. z p = <. y , z >. /\ p 2nd x ) )
29	24, 28	bitri 264	. . . . . . . 8 |- ( p ( 2nd |` ( _V X. _V ) ) x <-> ( E. y E. z p = <. y , z >. /\ p 2nd x ) )
30	29	anbi1i 731	. . . . . . 7 |- ( ( p ( 2nd |` ( _V X. _V ) ) x /\ p e. A ) <-> ( ( E. y E. z p = <. y , z >. /\ p 2nd x ) /\ p e. A ) )
31		19.41vv 1915	. . . . . . 7 |- ( E. y E. z ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) <-> ( E. y E. z p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) )
32	23, 30, 31	3bitr4i 292	. . . . . 6 |- ( ( p ( 2nd |` ( _V X. _V ) ) x /\ p e. A ) <-> E. y E. z ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) )
33	22, 32	bitri 264	. . . . 5 |- ( ( p e. A /\ p ( 2nd |` ( _V X. _V ) ) x ) <-> E. y E. z ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) )
34	33	exbii 1774	. . . 4 |- ( E. p ( p e. A /\ p ( 2nd |` ( _V X. _V ) ) x ) <-> E. p E. y E. z ( p = <. y , z >. /\ ( p 2nd x /\ p e. A ) ) )
35	1, 21, 34	3bitr4i 292	. . 3 |- ( E. y <. y , x >. e. A <-> E. p ( p e. A /\ p ( 2nd |` ( _V X. _V ) ) x ) )
36	16	elrn2 5365	. . 3 |- ( x e. ran A <-> E. y <. y , x >. e. A )
37	16	elima2 5472	. . 3 |- ( x e. ( ( 2nd |` ( _V X. _V ) ) " A ) <-> E. p ( p e. A /\ p ( 2nd |` ( _V X. _V ) ) x ) )
38	35, 36, 37	3bitr4i 292	. 2 |- ( x e. ran A <-> x e. ( ( 2nd |` ( _V X. _V ) ) " A ) )
39	38	eqriv 2619	1 |- ran A = ( ( 2nd |` ( _V X. _V ) ) " A )

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117   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-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-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-2nd 7169

This theorem is referenced by:  brrange  32041