Theorem isushgrm 15712

Description: The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
isushgrm	|- ( G e. U -> ( G e. USHGraph <-> E : dom E -1-1-> { s e. ~P V | E. j j e. s } ) )

Distinct variable groups:   j, s   V, s

Allowed substitution hints:   U( j, s)   E( j, s)   G( j, s)   V( j)

Proof of Theorem isushgrm

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

Step	Hyp	Ref	Expression
1		df-ushgrm 15710	. . 3 |- USHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } }
2	1	eleq2i 2273	. 2 |- ( G e. USHGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } } )
3		fveq2 5583	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isuhgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2257	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4885	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2210	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4884	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2255	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5583	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isuhgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2257	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3622	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2767	. . . 4 |- ( h = G -> { s e. ~P (Vtx ` h ) | E. j j e. s } = { s e. ~P V | E. j j e. s } )
15	5, 9, 14	f1eq123d 5521	. . 3 |- ( h = G -> ( (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } <-> E : dom E -1-1-> { s e. ~P V | E. j j e. s } ) )
16		vtxex 15661	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2777	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5583	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 15662	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2777	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5583	. . . . . . 7 |- ( g = h -> (iEdg ` g ) = (iEdg ` h ) )
24	23	adantr 276	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) = (iEdg ` h ) )
25		simpr 110	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> e = (iEdg ` h ) )
26	25	dmeqd 4885	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
27		simpr 110	. . . . . . . . . 10 |- ( ( g = h /\ v = (Vtx ` h ) ) -> v = (Vtx ` h ) )
28	27	pweqd 3622	. . . . . . . . 9 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ~P v = ~P (Vtx ` h ) )
29	28	rabeqdv 2767	. . . . . . . 8 |- ( ( g = h /\ v = (Vtx ` h ) ) -> { s e. ~P v | E. j j e. s } = { s e. ~P (Vtx ` h ) | E. j j e. s } )
30	29	adantr 276	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> { s e. ~P v | E. j j e. s } = { s e. ~P (Vtx ` h ) | E. j j e. s } )
31	25, 26, 30	f1eq123d 5521	. . . . . 6 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( e : dom e -1-1-> { s e. ~P v | E. j j e. s } <-> (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } ) )
32	22, 24, 31	sbcied2 3037	. . . . 5 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } <-> (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } ) )
33	18, 19, 32	sbcied2 3037	. . . 4 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } <-> (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } ) )
34	33	cbvabv 2331	. . 3 |- { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } } = { h | (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } }
35	15, 34	elab2g 2921	. 2 |- ( G e. U -> ( G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } } <-> E : dom E -1-1-> { s e. ~P V | E. j j e. s } ) )
36	2, 35	bitrid 192	1 |- ( G e. U -> ( G e. USHGraph <-> E : dom E -1-1-> { s e. ~P V | E. j j e. s } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2177   {cab 2192   {crab 2489   _Vcvv 2773   [.wsbc 2999   ~Pcpw 3617   dom cdm 4679   -1-1->wf1 5273   ` cfv 5276  Vtxcvtx 15655  iEdgciedg 15656  USHGraphcushgr 15708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-sub 8252  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-9 9109  df-n0 9303  df-dec 9512  df-ndx 12879  df-slot 12880  df-base 12882  df-edgf 15648  df-vtx 15657  df-iedg 15658  df-ushgrm 15710

This theorem is referenced by:  ushgrfm  15714