Theorem upgrspanop 16438

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 16434	. . . 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 16173	. . 3 |- ( G e. UPGraph -> (Vtx ` G ) e. _V )
12	1, 11	eqeltrid 2325	. 2 |- ( G e. UPGraph -> V e. _V )
13		iedgex 16174	. . . 4 |- ( G e. UPGraph -> (iEdg ` G ) e. _V )
14	2, 13	eqeltrid 2325	. . 3 |- ( G e. UPGraph -> E e. _V )
15		resexg 5098	. . 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 16204	1 |- ( G e. UPGraph -> <. V , ( E |` A ) >. e. UPGraph )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3708   |` cres 4771   ` cfv 5372  Vtxcvtx 16167  iEdgciedg 16168  UPGraphcupgr 16246

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-en 7013  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-uhgrm 16224  df-upgren 16248  df-subgr 16409

This theorem is referenced by: (None)