Theorem usgrspanop 16406

Description: A spanning subgraph of a simple graph represented by an ordered pair is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
usgrspanop	|- ( G e. USGraph -> <. V , ( E |` A ) >. e. USGraph )

Proof of Theorem usgrspanop

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uhgrspanop.v	. . . . 5 |- V = (Vtx ` G )
2		uhgrspanop.e	. . . . 5 |- E = (iEdg ` G )
3		vex 2818	. . . . . 6 |- g e. _V
4	3	a1i 9	. . . . 5 |- ( ( G e. USGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> g e. _V )
5		simprl 531	. . . . 5 |- ( ( G e. USGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> (Vtx ` g ) = V )
6		simprr 533	. . . . 5 |- ( ( G e. USGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> (iEdg ` g ) = ( E |` A ) )
7		simpl 109	. . . . 5 |- ( ( G e. USGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> G e. USGraph )
8	1, 2, 4, 5, 6, 7	usgrspan 16402	. . . 4 |- ( ( G e. USGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> g e. USGraph )
9	8	ex 115	. . 3 |- ( G e. USGraph -> ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) -> g e. USGraph ) )
10	9	alrimiv 1923	. 2 |- ( G e. USGraph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) -> g e. USGraph ) )
11		vtxex 16139	. . 3 |- ( G e. USGraph -> (Vtx ` G ) e. _V )
12	1, 11	eqeltrid 2321	. 2 |- ( G e. USGraph -> V e. _V )
13		iedgex 16140	. . . 4 |- ( G e. USGraph -> (iEdg ` G ) e. _V )
14	2, 13	eqeltrid 2321	. . 3 |- ( G e. USGraph -> E e. _V )
15		resexg 5083	. . 3 |- ( E e. _V -> ( E |` A ) e. _V )
16	14, 15	syl 14	. 2 |- ( G e. USGraph -> ( E |` A ) e. _V )
17	10, 12, 16	gropeld 16170	1 |- ( G e. USGraph -> <. V , ( E |` A ) >. e. USGraph )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   <.cop 3697   |` cres 4756   ` cfv 5357  Vtxcvtx 16133  iEdgciedg 16134  USGraphcusgr 16275

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-2o 6661  df-en 6989  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uhgrm 16190  df-upgren 16214  df-umgren 16215  df-uspgren 16276  df-usgren 16277  df-subgr 16375

This theorem is referenced by: (None)