Theorem isuhgr 32671

Description: The predicate "is an undirected hypergraph." (Contributed by AV, 18-Jan-2020.)

Hypotheses

Ref	Expression
isuhgr.v	|- V = ( Base ` G )
isuhgr.e	|- E = ( .ef ` 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 32669	. . 3 |- UHGraph = { g | [. ( Base ` g ) / v ]. [. ( .ef ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) }
2	1	eleq2i 2535	. 2 |- ( G e. UHGraph <-> G e. { g | [. ( Base ` g ) / v ]. [. ( .ef ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) } )
3		fveq2 5872	. . . . 5 |- ( h = G -> ( .ef ` h ) = ( .ef ` G ) )
4		isuhgr.e	. . . . 5 |- E = ( .ef ` G )
5	3, 4	syl6eqr 2516	. . . 4 |- ( h = G -> ( .ef ` h ) = E )
6	3	dmeqd 5215	. . . . 5 |- ( h = G -> dom ( .ef ` h ) = dom ( .ef ` G ) )
7	4	eqcomi 2470	. . . . . 6 |- ( .ef ` G ) = E
8	7	dmeqi 5214	. . . . 5 |- dom ( .ef ` G ) = dom E
9	6, 8	syl6eq 2514	. . . 4 |- ( h = G -> dom ( .ef ` h ) = dom E )
10		fveq2 5872	. . . . . . 7 |- ( h = G -> ( Base ` h ) = ( Base ` G ) )
11		isuhgr.v	. . . . . . 7 |- V = ( Base ` G )
12	10, 11	syl6eqr 2516	. . . . . 6 |- ( h = G -> ( Base ` h ) = V )
13	12	pweqd 4020	. . . . 5 |- ( h = G -> ~P ( Base ` h ) = ~P V )
14	13	difeq1d 3617	. . . 4 |- ( h = G -> ( ~P ( Base ` h ) \ { (/) } ) = ( ~P V \ { (/) } ) )
15	5, 9, 14	feq123d 5727	. . 3 |- ( h = G -> ( ( .ef ` h ) : dom ( .ef ` h ) --> ( ~P ( Base ` h ) \ { (/) } ) <-> E : dom E --> ( ~P V \ { (/) } ) ) )
16		fvex 5882	. . . . . 6 |- ( Base ` g ) e. _V
17	16	a1i 11	. . . . 5 |- ( g = h -> ( Base ` g ) e. _V )
18		fveq2 5872	. . . . 5 |- ( g = h -> ( Base ` g ) = ( Base ` h ) )
19		fvex 5882	. . . . . . 7 |- ( .ef ` g ) e. _V
20	19	a1i 11	. . . . . 6 |- ( ( g = h /\ v = ( Base ` h ) ) -> ( .ef ` g ) e. _V )
21		fveq2 5872	. . . . . . 7 |- ( g = h -> ( .ef ` g ) = ( .ef ` h ) )
22	21	adantr 465	. . . . . 6 |- ( ( g = h /\ v = ( Base ` h ) ) -> ( .ef ` g ) = ( .ef ` h ) )
23		simpr 461	. . . . . . 7 |- ( ( ( g = h /\ v = ( Base ` h ) ) /\ e = ( .ef ` h ) ) -> e = ( .ef ` h ) )
24	23	dmeqd 5215	. . . . . . 7 |- ( ( ( g = h /\ v = ( Base ` h ) ) /\ e = ( .ef ` h ) ) -> dom e = dom ( .ef ` h ) )
25		simpr 461	. . . . . . . . . 10 |- ( ( g = h /\ v = ( Base ` h ) ) -> v = ( Base ` h ) )
26	25	pweqd 4020	. . . . . . . . 9 |- ( ( g = h /\ v = ( Base ` h ) ) -> ~P v = ~P ( Base ` h ) )
27	26	difeq1d 3617	. . . . . . . 8 |- ( ( g = h /\ v = ( Base ` h ) ) -> ( ~P v \ { (/) } ) = ( ~P ( Base ` h ) \ { (/) } ) )
28	27	adantr 465	. . . . . . 7 |- ( ( ( g = h /\ v = ( Base ` h ) ) /\ e = ( .ef ` h ) ) -> ( ~P v \ { (/) } ) = ( ~P ( Base ` h ) \ { (/) } ) )
29	23, 24, 28	feq123d 5727	. . . . . 6 |- ( ( ( g = h /\ v = ( Base ` h ) ) /\ e = ( .ef ` h ) ) -> ( e : dom e --> ( ~P v \ { (/) } ) <-> ( .ef ` h ) : dom ( .ef ` h ) --> ( ~P ( Base ` h ) \ { (/) } ) ) )
30	20, 22, 29	sbcied2 3365	. . . . 5 |- ( ( g = h /\ v = ( Base ` h ) ) -> ( [. ( .ef ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) <-> ( .ef ` h ) : dom ( .ef ` h ) --> ( ~P ( Base ` h ) \ { (/) } ) ) )
31	17, 18, 30	sbcied2 3365	. . . 4 |- ( g = h -> ( [. ( Base ` g ) / v ]. [. ( .ef ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) <-> ( .ef ` h ) : dom ( .ef ` h ) --> ( ~P ( Base ` h ) \ { (/) } ) ) )
32	31	cbvabv 2600	. . 3 |- { g | [. ( Base ` g ) / v ]. [. ( .ef ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) } = { h | ( .ef ` h ) : dom ( .ef ` h ) --> ( ~P ( Base ` h ) \ { (/) } ) }
33	15, 32	elab2g 3248	. 2 |- ( G e. U -> ( G e. { g | [. ( Base ` g ) / v ]. [. ( .ef ` g ) / e ]. e : dom e --> ( ~P v \ { (/) } ) } <-> E : dom E --> ( ~P V \ { (/) } ) ) )
34	2, 33	syl5bb 257	1 |- ( G e. U -> ( G e. UHGraph <-> E : dom E --> ( ~P V \ { (/) } ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   {cab 2442   _Vcvv 3109   [.wsbc 3327   \ cdif 3468   (/)c0 3793   ~Pcpw 4015   {csn 4032   dom cdm 5008   -->wf 5590   ` cfv 5594   Basecbs 14735   .ef cedgf 32665   UHGraph cuhgr 32666

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-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586

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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  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-uni 4252  df-br 4457  df-opab 4516  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-uhgr 32669

This theorem is referenced by:  uhgf  32673  uhgeq12g  32675  ushguhg  32676  uhgrauhg  32678  uhg0e  32681  uhgrepe  32683