Theorem upgrunop 15977

Description: The union of two pseudographs (with the same vertex set): If <. V , E >. and <. V , F >. are pseudographs, then <. V , E u. F >. is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

Ref	Expression
upgrun.g	|- ( ph -> G e. UPGraph )
upgrun.h	|- ( ph -> H e. UPGraph )
upgrun.e	|- E = (iEdg ` G )
upgrun.f	|- F = (iEdg ` H )
upgrun.vg	|- V = (Vtx ` G )
upgrun.vh	|- ( ph -> (Vtx ` H ) = V )
upgrun.i	|- ( ph -> ( dom E i^i dom F ) = (/) )

Assertion

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

Proof of Theorem upgrunop

Step	Hyp	Ref	Expression
1		upgrun.g	. 2 |- ( ph -> G e. UPGraph )
2		upgrun.h	. 2 |- ( ph -> H e. UPGraph )
3		upgrun.e	. 2 |- E = (iEdg ` G )
4		upgrun.f	. 2 |- F = (iEdg ` H )
5		upgrun.vg	. 2 |- V = (Vtx ` G )
6		upgrun.vh	. 2 |- ( ph -> (Vtx ` H ) = V )
7		upgrun.i	. 2 |- ( ph -> ( dom E i^i dom F ) = (/) )
8		vtxex 15868	. . . . 5 |- ( G e. UPGraph -> (Vtx ` G ) e. _V )
9	1, 8	syl 14	. . . 4 |- ( ph -> (Vtx ` G ) e. _V )
10	5, 9	eqeltrid 2318	. . 3 |- ( ph -> V e. _V )
11		iedgex 15869	. . . . . 6 |- ( G e. UPGraph -> (iEdg ` G ) e. _V )
12	1, 11	syl 14	. . . . 5 |- ( ph -> (iEdg ` G ) e. _V )
13	3, 12	eqeltrid 2318	. . . 4 |- ( ph -> E e. _V )
14		iedgex 15869	. . . . . 6 |- ( H e. UPGraph -> (iEdg ` H ) e. _V )
15	2, 14	syl 14	. . . . 5 |- ( ph -> (iEdg ` H ) e. _V )
16	4, 15	eqeltrid 2318	. . . 4 |- ( ph -> F e. _V )
17		unexg 4540	. . . 4 |- ( ( E e. _V /\ F e. _V ) -> ( E u. F ) e. _V )
18	13, 16, 17	syl2anc 411	. . 3 |- ( ph -> ( E u. F ) e. _V )
19		opexg 4320	. . 3 |- ( ( V e. _V /\ ( E u. F ) e. _V ) -> <. V , ( E u. F ) >. e. _V )
20	10, 18, 19	syl2anc 411	. 2 |- ( ph -> <. V , ( E u. F ) >. e. _V )
21		opvtxfv 15872	. . 3 |- ( ( V e. _V /\ ( E u. F ) e. _V ) -> (Vtx ` <. V , ( E u. F ) >. ) = V )
22	10, 18, 21	syl2anc 411	. 2 |- ( ph -> (Vtx ` <. V , ( E u. F ) >. ) = V )
23		opiedgfv 15875	. . 3 |- ( ( V e. _V /\ ( E u. F ) e. _V ) -> (iEdg ` <. V , ( E u. F ) >. ) = ( E u. F ) )
24	10, 18, 23	syl2anc 411	. 2 |- ( ph -> (iEdg ` <. V , ( E u. F ) >. ) = ( E u. F ) )
25	1, 2, 3, 4, 5, 6, 7, 20, 22, 24	upgrun 15976	1 |- ( ph -> <. V , ( E u. F ) >. e. UPGraph )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   u. cun 3198   i^i cin 3199   (/)c0 3494   <.cop 3672   dom cdm 4725   ` cfv 5326  Vtxcvtx 15862  iEdgciedg 15863  UPGraphcupgr 15941

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-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-upgren 15943

This theorem is referenced by:  uspgrunop  16042