Theorem uspgrunop 16046

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 in 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 16035	. . 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 16035	. . 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	3, 6, 7, 8, 9, 10, 11	upgrunop 15981	1 |- ( ph -> <. V , ( E u. F ) >. e. UPGraph )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   u. cun 3198   i^i cin 3199   (/)c0 3494   <.cop 3672   dom cdm 4725   ` cfv 5326  Vtxcvtx 15866  iEdgciedg 15867  UPGraphcupgr 15945  USPGraphcuspgr 16007

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-upgren 15947  df-uspgren 16009

This theorem is referenced by: (None)