Theorem isupgr 25979

Description: The property of being an undirected pseudograph. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	|- V = (Vtx ` G )
isupgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
isupgr	|- ( G e. U -> ( G e. UPGraph <-> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )

Distinct variable groups:   x, G   x, V

Allowed substitution hints:   U( x)   E( x)

Proof of Theorem isupgr

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

Step	Hyp	Ref	Expression
1		df-upgr 25977	. . 3 |- UPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } }
2	1	eleq2i 2693	. 2 |- ( G e. UPGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } } )
3		fveq2 6191	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isupgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	syl6eqr 2674	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 5326	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2631	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 5325	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	syl6eq 2672	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 6191	. . . . . . . 8 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isupgr.v	. . . . . . . 8 |- V = (Vtx ` G )
12	10, 11	syl6eqr 2674	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 4163	. . . . . 6 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	difeq1d 3727	. . . . 5 |- ( h = G -> ( ~P (Vtx ` h ) \ { (/) } ) = ( ~P V \ { (/) } ) )
15	14	rabeqdv 3194	. . . 4 |- ( h = G -> { x e. ( ~P (Vtx ` h ) \ { (/) } ) | ( # ` x ) <_ 2 } = { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
16	5, 9, 15	feq123d 6034	. . 3 |- ( h = G -> ( (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ( ~P (Vtx ` h ) \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
17		fvexd 6203	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
18		fveq2 6191	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
19		fvexd 6203	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
20		fveq2 6191	. . . . . . 7 |- ( g = h -> (iEdg ` g ) = (iEdg ` h ) )
21	20	adantr 481	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) = (iEdg ` h ) )
22		simpr 477	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> e = (iEdg ` h ) )
23	22	dmeqd 5326	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
24		pweq 4161	. . . . . . . . . 10 |- ( v = (Vtx ` h ) -> ~P v = ~P (Vtx ` h ) )
25	24	ad2antlr 763	. . . . . . . . 9 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ~P v = ~P (Vtx ` h ) )
26	25	difeq1d 3727	. . . . . . . 8 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( ~P v \ { (/) } ) = ( ~P (Vtx ` h ) \ { (/) } ) )
27	26	rabeqdv 3194	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } = { x e. ( ~P (Vtx ` h ) \ { (/) } ) | ( # ` x ) <_ 2 } )
28	22, 23, 27	feq123d 6034	. . . . . 6 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ( ~P (Vtx ` h ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
29	19, 21, 28	sbcied2 3473	. . . . 5 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ( ~P (Vtx ` h ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
30	17, 18, 29	sbcied2 3473	. . . 4 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ( ~P (Vtx ` h ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
31	30	cbvabv 2747	. . 3 |- { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } } = { h | (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ( ~P (Vtx ` h ) \ { (/) } ) | ( # ` x ) <_ 2 } }
32	16, 31	elab2g 3353	. 2 |- ( G e. U -> ( G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } } <-> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
33	2, 32	syl5bb 272	1 |- ( G e. U -> ( G e. UPGraph <-> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975

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-nul 4789

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-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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  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-f 5892  df-fv 5896  df-upgr 25977

This theorem is referenced by:  wrdupgr  25980  upgrf  25981  upgrop  25989  umgrupgr  25998  upgr1e  26008  upgrun  26013  uspgrupgr  26071  subupgr  26179  upgrres  26198  upgrres1  26205