Theorem usgrun 16043

Description: The union U of two simple graphs G and H with the same vertex set V is a multigraph (not necessarily a simple graph!) with the vertex V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)

Hypotheses

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

Assertion

Ref	Expression
usgrun	|- ( ph -> U e. UMGraph )

Proof of Theorem usgrun

Step	Hyp	Ref	Expression
1		usgrun.g	. . 3 |- ( ph -> G e. USGraph )
2		usgrumgr 16034	. . 3 |- ( G e. USGraph -> G e. UMGraph )
3	1, 2	syl 14	. 2 |- ( ph -> G e. UMGraph )
4		usgrun.h	. . 3 |- ( ph -> H e. USGraph )
5		usgrumgr 16034	. . 3 |- ( H e. USGraph -> H e. UMGraph )
6	4, 5	syl 14	. 2 |- ( ph -> H e. UMGraph )
7		usgrun.e	. 2 |- E = (iEdg ` G )
8		usgrun.f	. 2 |- F = (iEdg ` H )
9		usgrun.vg	. 2 |- V = (Vtx ` G )
10		usgrun.vh	. 2 |- ( ph -> (Vtx ` H ) = V )
11		usgrun.i	. 2 |- ( ph -> ( dom E i^i dom F ) = (/) )
12		usgrun.u	. 2 |- ( ph -> U e. W )
13		usgrun.v	. 2 |- ( ph -> (Vtx ` U ) = V )
14		usgrun.un	. 2 |- ( ph -> (iEdg ` U ) = ( E u. F ) )
15	3, 6, 7, 8, 9, 10, 11, 12, 13, 14	umgrun 15978	1 |- ( ph -> U e. UMGraph )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   u. cun 3198   i^i cin 3199   (/)c0 3494   dom cdm 4725   ` cfv 5326  Vtxcvtx 15862  iEdgciedg 15863  UMGraphcumgr 15942  USGraphcusgr 16004

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-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

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-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-umgren 15944  df-usgren 16006

This theorem is referenced by: (None)