Theorem uspgrunop 26081

Description: The union of two simple pseudographs (with the same vertex set): If <. V , E >. and <. V , F >. are simple pseudographs, then <. V , E u. F >. is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 24-Oct-2021.)

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 ) = (/) )

Assertion

Ref	Expression
uspgrunop	|- ( ph -> <. V , ( E u. F ) >. e. UPGraph )

Proof of Theorem uspgrunop

Step	Hyp	Ref	Expression
1		uspgrun.g	. . 3 |- ( ph -> G e. USPGraph )
2		uspgrupgr 26071	. . 3 |- ( G e. USPGraph -> G e. UPGraph )
3	1, 2	syl 17	. 2 |- ( ph -> G e. UPGraph )
4		uspgrun.h	. . 3 |- ( ph -> H e. USPGraph )
5		uspgrupgr 26071	. . 3 |- ( H e. USPGraph -> H e. UPGraph )
6	4, 5	syl 17	. 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	3, 6, 7, 8, 9, 10, 11	upgrunop 26014	1 |- ( ph -> <. V , ( E u. F ) >. e. UPGraph )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   (/)c0 3915   <.cop 4183   dom cdm 5114   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975   USPGraph cuspgr 26043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877  df-upgr 25977  df-uspgr 26045

This theorem is referenced by: (None)