Theorem edgvalOLD 25942

Description: Obsolete version of edgval 25941 as of 8-Dec-2021. (Contributed by AV, 1-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
edgvalOLD	|- ( G e. V -> (Edg ` G ) = ran (iEdg ` G ) )

Proof of Theorem edgvalOLD

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-edg 25940	. . 3 |- Edg = ( g e. _V |-> ran (iEdg ` g ) )
2	1	a1i 11	. 2 |- ( G e. V -> Edg = ( g e. _V |-> ran (iEdg ` g ) ) )
3		fveq2 6191	. . . 4 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
4	3	rneqd 5353	. . 3 |- ( g = G -> ran (iEdg ` g ) = ran (iEdg ` G ) )
5	4	adantl 482	. 2 |- ( ( G e. V /\ g = G ) -> ran (iEdg ` g ) = ran (iEdg ` G ) )
6		elex 3212	. 2 |- ( G e. V -> G e. _V )
7		fvex 6201	. . . 4 |- (iEdg ` G ) e. _V
8	7	rnex 7100	. . 3 |- ran (iEdg ` G ) e. _V
9	8	a1i 11	. 2 |- ( G e. V -> ran (iEdg ` G ) e. _V )
10	2, 5, 6, 9	fvmptd 6288	1 |- ( G e. V -> (Edg ` G ) = ran (iEdg ` G ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ran crn 5115   ` 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-fv 5896  df-edg 25940

This theorem is referenced by:  edgiedgbOLD  25948  edg0iedg0OLD  25950  edginwlkOLD  26531