Theorem upgrunop 16282

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 16173	. . . . 5 |- ( G e. UPGraph -> (Vtx ` G ) e. _V )
9	1, 8	syl 14	. . . 4 |- ( ph -> (Vtx ` G ) e. _V )
10	5, 9	eqeltrid 2325	. . 3 |- ( ph -> V e. _V )
11		iedgex 16174	. . . . . 6 |- ( G e. UPGraph -> (iEdg ` G ) e. _V )
12	1, 11	syl 14	. . . . 5 |- ( ph -> (iEdg ` G ) e. _V )
13	3, 12	eqeltrid 2325	. . . 4 |- ( ph -> E e. _V )
14		iedgex 16174	. . . . . 6 |- ( H e. UPGraph -> (iEdg ` H ) e. _V )
15	2, 14	syl 14	. . . . 5 |- ( ph -> (iEdg ` H ) e. _V )
16	4, 15	eqeltrid 2325	. . . 4 |- ( ph -> F e. _V )
17		unexg 4584	. . . 4 |- ( ( E e. _V /\ F e. _V ) -> ( E u. F ) e. _V )
18	13, 16, 17	syl2anc 415	. . 3 |- ( ph -> ( E u. F ) e. _V )
19		opexg 4363	. . 3 |- ( ( V e. _V /\ ( E u. F ) e. _V ) -> <. V , ( E u. F ) >. e. _V )
20	10, 18, 19	syl2anc 415	. 2 |- ( ph -> <. V , ( E u. F ) >. e. _V )
21		opvtxfv 16177	. . 3 |- ( ( V e. _V /\ ( E u. F ) e. _V ) -> (Vtx ` <. V , ( E u. F ) >. ) = V )
22	10, 18, 21	syl2anc 415	. 2 |- ( ph -> (Vtx ` <. V , ( E u. F ) >. ) = V )
23		opiedgfv 16180	. . 3 |- ( ( V e. _V /\ ( E u. F ) e. _V ) -> (iEdg ` <. V , ( E u. F ) >. ) = ( E u. F ) )
24	10, 18, 23	syl2anc 415	. 2 |- ( ph -> (iEdg ` <. V , ( E u. F ) >. ) = ( E u. F ) )
25	1, 2, 3, 4, 5, 6, 7, 20, 22, 24	upgrun 16281	1 |- ( ph -> <. V , ( E u. F ) >. e. UPGraph )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   <.cop 3708   dom cdm 4769   ` cfv 5372  Vtxcvtx 16167  iEdgciedg 16168  UPGraphcupgr 16246

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-upgren 16248

This theorem is referenced by:  uspgrunop  16347