Theorem isuslgra 24470

Description: The property of being an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)

Assertion

Ref	Expression
isuslgra	|- ( ( V e. W /\ E e. X ) -> ( V USLGrph E <-> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )

Distinct variable groups:   x, E   x, V

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

Proof of Theorem isuslgra

Dummy variables v e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1eq1 5782	. . . 4 |- ( e = E -> ( e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } ) )
2	1	adantl 466	. . 3 |- ( ( v = V /\ e = E ) -> ( e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } ) )
3		dmeq 5213	. . . . 5 |- ( e = E -> dom e = dom E )
4	3	adantl 466	. . . 4 |- ( ( v = V /\ e = E ) -> dom e = dom E )
5		f1eq2 5783	. . . 4 |- ( dom e = dom E -> ( E : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : dom E -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } ) )
6	4, 5	syl 16	. . 3 |- ( ( v = V /\ e = E ) -> ( E : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : dom E -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } ) )
7		simpl 457	. . . . . 6 |- ( ( v = V /\ e = E ) -> v = V )
8	7	pweqd 4020	. . . . 5 |- ( ( v = V /\ e = E ) -> ~P v = ~P V )
9	8	difeq1d 3617	. . . 4 |- ( ( v = V /\ e = E ) -> ( ~P v \ { (/) } ) = ( ~P V \ { (/) } ) )
10		rabeq 3103	. . . 4 |- ( ( ~P v \ { (/) } ) = ( ~P V \ { (/) } ) -> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } = { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
11		f1eq3 5784	. . . 4 |- ( { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } = { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } -> ( E : dom E -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
12	9, 10, 11	3syl 20	. . 3 |- ( ( v = V /\ e = E ) -> ( E : dom E -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
13	2, 6, 12	3bitrd 279	. 2 |- ( ( v = V /\ e = E ) -> ( e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
14		df-uslgra 24459	. 2 |- USLGrph = { <. v , e >. | e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } }
15	13, 14	brabga 4770	1 |- ( ( V e. W /\ E e. X ) -> ( V USLGrph E <-> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   {crab 2811   \ cdif 3468   (/)c0 3793   ~Pcpw 4015   {csn 4032   class class class wbr 4456   dom cdm 5008   -1-1->wf1 5591   ` cfv 5594   <_ cle 9646   2c2 10606   #chash 12408   USLGrph cuslg 24456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-uslgra 24459

This theorem is referenced by:  uslgraf  24472  uslisushgra  24490  uslisumgra  24491  usisuslgra  24492  uslgra1  24499