Theorem funcnv3 5959

Description: A condition showing a class is single-rooted. (See funcnv 5958). (Contributed by NM, 26-May-2006.)

Assertion

Ref	Expression
funcnv3	|- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y )

Distinct variable group:   x, y, A

Proof of Theorem funcnv3

Step	Hyp	Ref	Expression
1		dfrn2 5311	. . . . . 6 |- ran A = { y | E. x x A y }
2	1	abeq2i 2735	. . . . 5 |- ( y e. ran A <-> E. x x A y )
3	2	biimpi 206	. . . 4 |- ( y e. ran A -> E. x x A y )
4	3	biantrurd 529	. . 3 |- ( y e. ran A -> ( E* x x A y <-> ( E. x x A y /\ E* x x A y ) ) )
5	4	ralbiia 2979	. 2 |- ( A. y e. ran A E* x x A y <-> A. y e. ran A ( E. x x A y /\ E* x x A y ) )
6		funcnv 5958	. 2 |- ( Fun `' A <-> A. y e. ran A E* x x A y )
7		df-reu 2919	. . . 4 |- ( E! x e. dom A x A y <-> E! x ( x e. dom A /\ x A y ) )
8		vex 3203	. . . . . . 7 |- x e. _V
9		vex 3203	. . . . . . 7 |- y e. _V
10	8, 9	breldm 5329	. . . . . 6 |- ( x A y -> x e. dom A )
11	10	pm4.71ri 665	. . . . 5 |- ( x A y <-> ( x e. dom A /\ x A y ) )
12	11	eubii 2492	. . . 4 |- ( E! x x A y <-> E! x ( x e. dom A /\ x A y ) )
13		eu5 2496	. . . 4 |- ( E! x x A y <-> ( E. x x A y /\ E* x x A y ) )
14	7, 12, 13	3bitr2i 288	. . 3 |- ( E! x e. dom A x A y <-> ( E. x x A y /\ E* x x A y ) )
15	14	ralbii 2980	. 2 |- ( A. y e. ran A E! x e. dom A x A y <-> A. y e. ran A ( E. x x A y /\ E* x x A y ) )
16	5, 6, 15	3bitr4i 292	1 |- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   A.wral 2912   E!wreu 2914   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   Fun wfun 5882

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-reu 2919  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890

This theorem is referenced by: (None)