Theorem upgrspanop 16524

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 2824	. . . . . 6 |- g e. _V
4	3	a1i 9	. . . . 5 |- ( ( G e. UPGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> g e. _V )
5		simprl 535	. . . . 5 |- ( ( G e. UPGraph /\ ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) ) -> (Vtx ` g ) = V )
6		simprr 537	. . . . 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 16520	. . . 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 1927	. 2 |- ( G e. UPGraph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = ( E |` A ) ) -> g e. UPGraph ) )
11		vtxex 16259	. . 3 |- ( G e. UPGraph -> (Vtx ` G ) e. _V )
12	1, 11	eqeltrid 2325	. 2 |- ( G e. UPGraph -> V e. _V )
13		iedgex 16260	. . . 4 |- ( G e. UPGraph -> (iEdg ` G ) e. _V )
14	2, 13	eqeltrid 2325	. . 3 |- ( G e. UPGraph -> E e. _V )
15		resexg 5103	. . 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 16290	1 |- ( G e. UPGraph -> <. V , ( E |` A ) >. e. UPGraph )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3712   |` cres 4776   ` cfv 5377  Vtxcvtx 16253  iEdgciedg 16254  UPGraphcupgr 16332

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-en 7023  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-upgren 16334  df-subgr 16495

This theorem is used by: (None)