Theorem usgrspanop 16155

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 2805	. . . . . 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 16151	. . . 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 1922	. 2 |- ( G e. USGraph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) -> g e. USGraph ) )
11		vtxex 15888	. . 3 |- ( G e. USGraph -> (Vtx ` G ) e. _V )
12	1, 11	eqeltrid 2318	. 2 |- ( G e. USGraph -> V e. _V )
13		iedgex 15889	. . . 4 |- ( G e. USGraph -> (iEdg ` G ) e. _V )
14	2, 13	eqeltrid 2318	. . 3 |- ( G e. USGraph -> E e. _V )
15		resexg 5053	. . 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 15919	1 |- ( G e. USGraph -> <. V , ( E |` A ) >. e. USGraph )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   <.cop 3672   |` cres 4727   ` cfv 5326  Vtxcvtx 15882  iEdgciedg 15883  USGraphcusgr 16024

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-1o 6582  df-2o 6583  df-en 6910  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13103  df-slot 13104  df-base 13106  df-edgf 15875  df-vtx 15884  df-iedg 15885  df-edg 15928  df-uhgrm 15939  df-upgren 15963  df-umgren 15964  df-uspgren 16025  df-usgren 16026  df-subgr 16124

This theorem is referenced by: (None)