Theorem uspgrun 16203

Description: The union U of two simple pseudographs G and H with the same vertex set V is a pseudograph with the vertex V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 16-Oct-2020.)

Hypotheses

Ref	Expression
uspgrun.g	|- ( ph -> G e. USPGraph )
uspgrun.h	|- ( ph -> H e. USPGraph )
uspgrun.e	|- E = (iEdg ` G )
uspgrun.f	|- F = (iEdg ` H )
uspgrun.vg	|- V = (Vtx ` G )
uspgrun.vh	|- ( ph -> (Vtx ` H ) = V )
uspgrun.i	|- ( ph -> ( dom E i^i dom F ) = (/) )
uspgrun.u	|- ( ph -> U e. W )
uspgrun.v	|- ( ph -> (Vtx ` U ) = V )
uspgrun.un	|- ( ph -> (iEdg ` U ) = ( E u. F ) )

Assertion

Ref	Expression
uspgrun	|- ( ph -> U e. UPGraph )

Proof of Theorem uspgrun

Step	Hyp	Ref	Expression
1		uspgrun.g	. . 3 |- ( ph -> G e. USPGraph )
2		uspgrupgr 16193	. . 3 |- ( G e. USPGraph -> G e. UPGraph )
3	1, 2	syl 14	. 2 |- ( ph -> G e. UPGraph )
4		uspgrun.h	. . 3 |- ( ph -> H e. USPGraph )
5		uspgrupgr 16193	. . 3 |- ( H e. USPGraph -> H e. UPGraph )
6	4, 5	syl 14	. 2 |- ( ph -> H e. UPGraph )
7		uspgrun.e	. 2 |- E = (iEdg ` G )
8		uspgrun.f	. 2 |- F = (iEdg ` H )
9		uspgrun.vg	. 2 |- V = (Vtx ` G )
10		uspgrun.vh	. 2 |- ( ph -> (Vtx ` H ) = V )
11		uspgrun.i	. 2 |- ( ph -> ( dom E i^i dom F ) = (/) )
12		uspgrun.u	. 2 |- ( ph -> U e. W )
13		uspgrun.v	. 2 |- ( ph -> (Vtx ` U ) = V )
14		uspgrun.un	. 2 |- ( ph -> (iEdg ` U ) = ( E u. F ) )
15	3, 6, 7, 8, 9, 10, 11, 12, 13, 14	upgrun 16138	1 |- ( ph -> U e. UPGraph )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   u. cun 3211   i^i cin 3212   (/)c0 3510   dom cdm 4751   ` cfv 5354  Vtxcvtx 16024  iEdgciedg 16025  UPGraphcupgr 16103  USPGraphcuspgr 16165

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 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-sub 8448  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-dec 9713  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-upgren 16105  df-uspgren 16167

This theorem is referenced by: (None)