Theorem edgiedgbOLD 25948

Description: Obsolete version of edgiedgb 25947 as of 8-Dec-2021. (Contributed by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
edgiedgb.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
edgiedgbOLD	|- ( ( G e. W /\ Fun I ) -> ( E e. (Edg ` G ) <-> E. x e. dom I E = ( I ` x ) ) )

Distinct variable groups:   x, E   x, I

Allowed substitution hints:   G( x)   W( x)

Proof of Theorem edgiedgbOLD

Step	Hyp	Ref	Expression
1		edgvalOLD 25942	. . . . 5 |- ( G e. W -> (Edg ` G ) = ran (iEdg ` G ) )
2	1	adantr 481	. . . 4 |- ( ( G e. W /\ Fun I ) -> (Edg ` G ) = ran (iEdg ` G ) )
3		edgiedgb.i	. . . . . . 7 |- I = (iEdg ` G )
4	3	eqcomi 2631	. . . . . 6 |- (iEdg ` G ) = I
5	4	a1i 11	. . . . 5 |- ( ( G e. W /\ Fun I ) -> (iEdg ` G ) = I )
6	5	rneqd 5353	. . . 4 |- ( ( G e. W /\ Fun I ) -> ran (iEdg ` G ) = ran I )
7	2, 6	eqtrd 2656	. . 3 |- ( ( G e. W /\ Fun I ) -> (Edg ` G ) = ran I )
8	7	eleq2d 2687	. 2 |- ( ( G e. W /\ Fun I ) -> ( E e. (Edg ` G ) <-> E e. ran I ) )
9		elrnrexdmb 6364	. . 3 |- ( Fun I -> ( E e. ran I <-> E. x e. dom I E = ( I ` x ) ) )
10	9	adantl 482	. 2 |- ( ( G e. W /\ Fun I ) -> ( E e. ran I <-> E. x e. dom I E = ( I ` x ) ) )
11	8, 10	bitrd 268	1 |- ( ( G e. W /\ Fun I ) -> ( E e. (Edg ` G ) <-> E. x e. dom I E = ( I ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   dom cdm 5114   ran crn 5115   Fun wfun 5882   ` cfv 5888  iEdgciedg 25875  Edgcedg 25939

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-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-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-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-fv 5896  df-edg 25940

This theorem is referenced by: (None)