Theorem upgrspanop 16410

Description: A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.)

Hypotheses

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

Assertion

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

Proof of Theorem upgrspanop

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. UPGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> g e. _V )
5		simprl 531	. . . . 5 |- ( ( G e. UPGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> (Vtx ` g ) = V )
6		simprr 533	. . . . 5 |- ( ( G e. UPGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> (iEdg ` g ) = ( E |` A ) )
7		simpl 109	. . . . 5 |- ( ( G e. UPGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> G e. UPGraph )
8	1, 2, 4, 5, 6, 7	upgrspan 16406	. . . 4 |- ( ( G e. UPGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> g e. UPGraph )
9	8	ex 115	. . 3 |- ( G e. UPGraph -> ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) -> g e. UPGraph ) )
10	9	alrimiv 1923	. 2 |- ( G e. UPGraph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) -> g e. UPGraph ) )
11		vtxex 16145	. . 3 |- ( G e. UPGraph -> (Vtx ` G ) e. _V )
12	1, 11	eqeltrid 2321	. 2 |- ( G e. UPGraph -> V e. _V )
13		iedgex 16146	. . . 4 |- ( G e. UPGraph -> (iEdg ` G ) e. _V )
14	2, 13	eqeltrid 2321	. . 3 |- ( G e. UPGraph -> E e. _V )
15		resexg 5085	. . 3 |- ( E e. _V -> ( E |` A ) e. _V )
16	14, 15	syl 14	. 2 |- ( G e. UPGraph -> ( E |` A ) e. _V )
17	10, 12, 16	gropeld 16176	1 |- ( G e. UPGraph -> <. V , ( E |` A ) >. e. UPGraph )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   <.cop 3698   |` cres 4758   ` cfv 5359  Vtxcvtx 16139  iEdgciedg 16140  UPGraphcupgr 16218

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-14 2208  ax-ext 2216  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-cnre 8256

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 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-iord 4493  df-on 4495  df-suc 4498  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-1o 6662  df-2o 6663  df-en 6991  df-sub 8465  df-inn 9260  df-2 9318  df-3 9319  df-4 9320  df-5 9321  df-6 9322  df-7 9323  df-8 9324  df-9 9325  df-n0 9519  df-dec 9733  df-ndx 13305  df-slot 13306  df-base 13308  df-edgf 16132  df-vtx 16141  df-iedg 16142  df-edg 16185  df-uhgrm 16196  df-upgren 16220  df-subgr 16381

This theorem is referenced by: (None)