Theorem isuhgr 39161

Description: The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
isuhgr	|- ( G e. U -> ( G e. UHGraph <-> E : dom E --> ( ~P V \ { (/) } ) ) )

Proof of Theorem isuhgr

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

Step	Hyp	Ref	Expression
1		df-uhgr 39159	. . 3 |- UHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) }
2	1	eleq2i 2523	. 2 |- ( G e. UHGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) } )
3		fveq2 5870	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isuhgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	syl6eqr 2505	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 5040	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2462	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 5039	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	syl6eq 2503	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5870	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isuhgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	syl6eqr 2505	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3958	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	difeq1d 3552	. . . 4 |- ( h = G -> ( ~P (Vtx ` h ) \ { (/) } ) = ( ~P V \ { (/) } ) )
15	5, 9, 14	feq123d 5723	. . 3 |- ( h = G -> ( (iEdg ` h ) : dom (iEdg ` h ) --> ( ~P (Vtx ` h ) \ { (/) } ) <-> E : dom E --> ( ~P V \ { (/) } ) ) )
16		fvex 5880	. . . . . 6 |- (Vtx ` g ) e. _V
17	16	a1i 11	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
18		fveq2 5870	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
19		fvex 5880	. . . . . . 7 |- (iEdg ` g ) e. _V
20	19	a1i 11	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
21		fveq2 5870	. . . . . . 7 |- ( g = h -> (iEdg ` g ) = (iEdg ` h ) )
22	21	adantr 467	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) = (iEdg ` h ) )
23		simpr 463	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> e = (iEdg ` h ) )
24	23	dmeqd 5040	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
25		simpr 463	. . . . . . . . . 10 |- ( ( g = h /\ v = (Vtx ` h ) ) -> v = (Vtx ` h ) )
26	25	pweqd 3958	. . . . . . . . 9 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ~P v = ~P (Vtx ` h ) )
27	26	difeq1d 3552	. . . . . . . 8 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( ~P v \ { (/) } ) = ( ~P (Vtx ` h ) \ { (/) } ) )
28	27	adantr 467	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( ~P v \ { (/) } ) = ( ~P (Vtx ` h ) \ { (/) } ) )
29	23, 24, 28	feq123d 5723	. . . . . 6 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( e : dom e --> ( ~P v \ { (/) } ) <-> (iEdg ` h ) : dom (iEdg ` h ) --> ( ~P (Vtx ` h ) \ { (/) } ) ) )
30	20, 22, 29	sbcied2 3307	. . . . 5 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (iEdg ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) <-> (iEdg ` h ) : dom (iEdg ` h ) --> ( ~P (Vtx ` h ) \ { (/) } ) ) )
31	17, 18, 30	sbcied2 3307	. . . 4 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) <-> (iEdg ` h ) : dom (iEdg ` h ) --> ( ~P (Vtx ` h ) \ { (/) } ) ) )
32	31	cbvabv 2577	. . 3 |- { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) } = { h | (iEdg ` h ) : dom (iEdg ` h ) --> ( ~P (Vtx ` h ) \ { (/) } ) }
33	15, 32	elab2g 3189	. 2 |- ( G e. U -> ( G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) } <-> E : dom E --> ( ~P V \ { (/) } ) ) )
34	2, 33	syl5bb 261	1 |- ( G e. U -> ( G e. UHGraph <-> E : dom E --> ( ~P V \ { (/) } ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   {cab 2439   _Vcvv 3047   [.wsbc 3269   \ cdif 3403   (/)c0 3733   ~Pcpw 3953   {csn 3970   dom cdm 4837   -->wf 5581   ` cfv 5585  Vtxcvtx 39111  iEdgciedg 39112   UHGraph cuhgr 39157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-nul 4537

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593  df-uhgr 39159

This theorem is referenced by:  uhgrf  39163  uhgreq12g  39166  ushgruhgr  39169  uhgrauhgr  39171  isuhgrop  39173  uhgr0e  39174  uhgr0  39176  uhgrun  39177  incistruhgr  39181  upgruhgr  39201  subuhgr  39368