Theorem vtxdfifiun 16103

Description: The degree of a vertex in the union of two pseudographs of finite size on the same finite vertex set is the sum of the degrees of the vertex in each pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 19-Feb-2021.)

Hypotheses

Ref	Expression
vtxdun.i	|- I = (iEdg ` G )
vtxdun.j	|- J = (iEdg ` H )
vtxdun.vg	|- V = (Vtx ` G )
vtxdun.vh	|- ( ph -> (Vtx ` H ) = V )
vtxdun.vu	|- ( ph -> (Vtx ` U ) = V )
vtxdfifiun.v	|- ( ph -> V e. Fin )
vtxdfifiun.g	|- ( ph -> G e. UPGraph )
vtxdfifiun.h	|- ( ph -> H e. UPGraph )
vtxdun.d	|- ( ph -> ( dom I i^i dom J ) = (/) )
vtxdun.fi	|- ( ph -> Fun I )
vtxdun.fj	|- ( ph -> Fun J )
vtxdun.n	|- ( ph -> N e. V )
vtxdun.u	|- ( ph -> (iEdg ` U ) = ( I u. J ) )
vtxdfiun.a	|- ( ph -> dom I e. Fin )
vtxdfiun.b	|- ( ph -> dom J e. Fin )

Assertion

Ref	Expression
vtxdfifiun	|- ( ph -> ( (VtxDeg ` U ) ` N ) = ( ( (VtxDeg ` G ) ` N ) + ( (VtxDeg ` H ) ` N ) ) )

Proof of Theorem vtxdfifiun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 2517	. . . . . . . 8 |- { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } = { x | ( x e. dom (iEdg ` U ) /\ N e. ( (iEdg ` U ) ` x ) ) }
2		vtxdun.u	. . . . . . . . . . . . . . 15 |- ( ph -> (iEdg ` U ) = ( I u. J ) )
3	2	dmeqd 4931	. . . . . . . . . . . . . 14 |- ( ph -> dom (iEdg ` U ) = dom ( I u. J ) )
4		dmun 4936	. . . . . . . . . . . . . 14 |- dom ( I u. J ) = ( dom I u. dom J )
5	3, 4	eqtrdi 2278	. . . . . . . . . . . . 13 |- ( ph -> dom (iEdg ` U ) = ( dom I u. dom J ) )
6	5	eleq2d 2299	. . . . . . . . . . . 12 |- ( ph -> ( x e. dom (iEdg ` U ) <-> x e. ( dom I u. dom J ) ) )
7		elun 3346	. . . . . . . . . . . 12 |- ( x e. ( dom I u. dom J ) <-> ( x e. dom I \/ x e. dom J ) )
8	6, 7	bitrdi 196	. . . . . . . . . . 11 |- ( ph -> ( x e. dom (iEdg ` U ) <-> ( x e. dom I \/ x e. dom J ) ) )
9	8	anbi1d 465	. . . . . . . . . 10 |- ( ph -> ( ( x e. dom (iEdg ` U ) /\ N e. ( (iEdg ` U ) ` x ) ) <-> ( ( x e. dom I \/ x e. dom J ) /\ N e. ( (iEdg ` U ) ` x ) ) ) )
10		andir 824	. . . . . . . . . 10 |- ( ( ( x e. dom I \/ x e. dom J ) /\ N e. ( (iEdg ` U ) ` x ) ) <-> ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) )
11	9, 10	bitrdi 196	. . . . . . . . 9 |- ( ph -> ( ( x e. dom (iEdg ` U ) /\ N e. ( (iEdg ` U ) ` x ) ) <-> ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) ) )
12	11	abbidv 2347	. . . . . . . 8 |- ( ph -> { x | ( x e. dom (iEdg ` U ) /\ N e. ( (iEdg ` U ) ` x ) ) } = { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) } )
13	1, 12	eqtrid 2274	. . . . . . 7 |- ( ph -> { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } = { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) } )
14		unab 3472	. . . . . . . . 9 |- ( { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } u. { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } ) = { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) }
15	14	eqcomi 2233	. . . . . . . 8 |- { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) } = ( { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } u. { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } )
16	15	a1i 9	. . . . . . 7 |- ( ph -> { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) } = ( { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } u. { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } ) )
17		df-rab 2517	. . . . . . . . 9 |- { x e. dom I | N e. ( (iEdg ` U ) ` x ) } = { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) }
18	2	fveq1d 5637	. . . . . . . . . . . . 13 |- ( ph -> ( (iEdg ` U ) ` x ) = ( ( I u. J ) ` x ) )
19	18	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom I ) -> ( (iEdg ` U ) ` x ) = ( ( I u. J ) ` x ) )
20		vtxdun.fi	. . . . . . . . . . . . . . 15 |- ( ph -> Fun I )
21	20	funfnd 5355	. . . . . . . . . . . . . 14 |- ( ph -> I Fn dom I )
22	21	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom I ) -> I Fn dom I )
23		vtxdun.fj	. . . . . . . . . . . . . . 15 |- ( ph -> Fun J )
24	23	funfnd 5355	. . . . . . . . . . . . . 14 |- ( ph -> J Fn dom J )
25	24	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom I ) -> J Fn dom J )
26		vtxdun.d	. . . . . . . . . . . . . 14 |- ( ph -> ( dom I i^i dom J ) = (/) )
27	26	anim1i 340	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom I ) -> ( ( dom I i^i dom J ) = (/) /\ x e. dom I ) )
28		fvun1 5708	. . . . . . . . . . . . 13 |- ( ( I Fn dom I /\ J Fn dom J /\ ( ( dom I i^i dom J ) = (/) /\ x e. dom I ) ) -> ( ( I u. J ) ` x ) = ( I ` x ) )
29	22, 25, 27, 28	syl3anc 1271	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom I ) -> ( ( I u. J ) ` x ) = ( I ` x ) )
30	19, 29	eqtrd 2262	. . . . . . . . . . 11 |- ( ( ph /\ x e. dom I ) -> ( (iEdg ` U ) ` x ) = ( I ` x ) )
31	30	eleq2d 2299	. . . . . . . . . 10 |- ( ( ph /\ x e. dom I ) -> ( N e. ( (iEdg ` U ) ` x ) <-> N e. ( I ` x ) ) )
32	31	rabbidva 2788	. . . . . . . . 9 |- ( ph -> { x e. dom I | N e. ( (iEdg ` U ) ` x ) } = { x e. dom I | N e. ( I ` x ) } )
33	17, 32	eqtr3id 2276	. . . . . . . 8 |- ( ph -> { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } = { x e. dom I | N e. ( I ` x ) } )
34		df-rab 2517	. . . . . . . . 9 |- { x e. dom J | N e. ( (iEdg ` U ) ` x ) } = { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) }
35	18	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom J ) -> ( (iEdg ` U ) ` x ) = ( ( I u. J ) ` x ) )
36	21	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom J ) -> I Fn dom I )
37	24	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom J ) -> J Fn dom J )
38	26	anim1i 340	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom J ) -> ( ( dom I i^i dom J ) = (/) /\ x e. dom J ) )
39		fvun2 5709	. . . . . . . . . . . . 13 |- ( ( I Fn dom I /\ J Fn dom J /\ ( ( dom I i^i dom J ) = (/) /\ x e. dom J ) ) -> ( ( I u. J ) ` x ) = ( J ` x ) )
40	36, 37, 38, 39	syl3anc 1271	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom J ) -> ( ( I u. J ) ` x ) = ( J ` x ) )
41	35, 40	eqtrd 2262	. . . . . . . . . . 11 |- ( ( ph /\ x e. dom J ) -> ( (iEdg ` U ) ` x ) = ( J ` x ) )
42	41	eleq2d 2299	. . . . . . . . . 10 |- ( ( ph /\ x e. dom J ) -> ( N e. ( (iEdg ` U ) ` x ) <-> N e. ( J ` x ) ) )
43	42	rabbidva 2788	. . . . . . . . 9 |- ( ph -> { x e. dom J | N e. ( (iEdg ` U ) ` x ) } = { x e. dom J | N e. ( J ` x ) } )
44	34, 43	eqtr3id 2276	. . . . . . . 8 |- ( ph -> { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } = { x e. dom J | N e. ( J ` x ) } )
45	33, 44	uneq12d 3360	. . . . . . 7 |- ( ph -> ( { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } u. { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } ) = ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) )
46	13, 16, 45	3eqtrd 2266	. . . . . 6 |- ( ph -> { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } = ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) )
47	46	fveq2d 5639	. . . . 5 |- ( ph -> (♯ ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) = (♯ ` ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) ) )
48		vtxdun.vg	. . . . . . 7 |- V = (Vtx ` G )
49		vtxdun.i	. . . . . . 7 |- I = (iEdg ` G )
50		eqid 2229	. . . . . . 7 |- dom I = dom I
51		vtxdfiun.a	. . . . . . 7 |- ( ph -> dom I e. Fin )
52		vtxdfifiun.v	. . . . . . 7 |- ( ph -> V e. Fin )
53		vtxdun.n	. . . . . . 7 |- ( ph -> N e. V )
54		vtxdfifiun.g	. . . . . . 7 |- ( ph -> G e. UPGraph )
55	48, 49, 50, 51, 52, 53, 54	vtxedgfi 16095	. . . . . 6 |- ( ph -> { x e. dom I | N e. ( I ` x ) } e. Fin )
56		eqid 2229	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
57		vtxdun.j	. . . . . . 7 |- J = (iEdg ` H )
58		eqid 2229	. . . . . . 7 |- dom J = dom J
59		vtxdfiun.b	. . . . . . 7 |- ( ph -> dom J e. Fin )
60		vtxdun.vh	. . . . . . . 8 |- ( ph -> (Vtx ` H ) = V )
61	60, 52	eqeltrd 2306	. . . . . . 7 |- ( ph -> (Vtx ` H ) e. Fin )
62	53, 60	eleqtrrd 2309	. . . . . . 7 |- ( ph -> N e. (Vtx ` H ) )
63		vtxdfifiun.h	. . . . . . 7 |- ( ph -> H e. UPGraph )
64	56, 57, 58, 59, 61, 62, 63	vtxedgfi 16095	. . . . . 6 |- ( ph -> { x e. dom J | N e. ( J ` x ) } e. Fin )
65		ssrab2 3310	. . . . . . . . 9 |- { x e. dom I | N e. ( I ` x ) } C_ dom I
66		ssrab2 3310	. . . . . . . . 9 |- { x e. dom J | N e. ( J ` x ) } C_ dom J
67		ss2in 3433	. . . . . . . . 9 |- ( ( { x e. dom I | N e. ( I ` x ) } C_ dom I /\ { x e. dom J | N e. ( J ` x ) } C_ dom J ) -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ ( dom I i^i dom J ) )
68	65, 66, 67	mp2an 426	. . . . . . . 8 |- ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ ( dom I i^i dom J )
69	68, 26	sseqtrid 3275	. . . . . . 7 |- ( ph -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ (/) )
70		ss0 3533	. . . . . . 7 |- ( ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ (/) -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) = (/) )
71	69, 70	syl 14	. . . . . 6 |- ( ph -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) = (/) )
72		hashun 11058	. . . . . 6 |- ( ( { x e. dom I | N e. ( I ` x ) } e. Fin /\ { x e. dom J | N e. ( J ` x ) } e. Fin /\ ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) = (/) ) -> (♯ ` ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) ) = ( (♯ ` { x e. dom I | N e. ( I ` x ) } ) + (♯ ` { x e. dom J | N e. ( J ` x ) } ) ) )
73	55, 64, 71, 72	syl3anc 1271	. . . . 5 |- ( ph -> (♯ ` ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) ) = ( (♯ ` { x e. dom I | N e. ( I ` x ) } ) + (♯ ` { x e. dom J | N e. ( J ` x ) } ) ) )
74	47, 73	eqtrd 2262	. . . 4 |- ( ph -> (♯ ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) = ( (♯ ` { x e. dom I | N e. ( I ` x ) } ) + (♯ ` { x e. dom J | N e. ( J ` x ) } ) ) )
75		df-rab 2517	. . . . . . . 8 |- { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom (iEdg ` U ) /\ ( (iEdg ` U ) ` x ) = { N } ) }
76	8	anbi1d 465	. . . . . . . . . 10 |- ( ph -> ( ( x e. dom (iEdg ` U ) /\ ( (iEdg ` U ) ` x ) = { N } ) <-> ( ( x e. dom I \/ x e. dom J ) /\ ( (iEdg ` U ) ` x ) = { N } ) ) )
77		andir 824	. . . . . . . . . 10 |- ( ( ( x e. dom I \/ x e. dom J ) /\ ( (iEdg ` U ) ` x ) = { N } ) <-> ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) )
78	76, 77	bitrdi 196	. . . . . . . . 9 |- ( ph -> ( ( x e. dom (iEdg ` U ) /\ ( (iEdg ` U ) ` x ) = { N } ) <-> ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) ) )
79	78	abbidv 2347	. . . . . . . 8 |- ( ph -> { x | ( x e. dom (iEdg ` U ) /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) } )
80	75, 79	eqtrid 2274	. . . . . . 7 |- ( ph -> { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } = { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) } )
81		unab 3472	. . . . . . . . 9 |- ( { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } u. { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } ) = { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) }
82	81	eqcomi 2233	. . . . . . . 8 |- { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) } = ( { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } u. { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } )
83	82	a1i 9	. . . . . . 7 |- ( ph -> { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) } = ( { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } u. { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } ) )
84		df-rab 2517	. . . . . . . . 9 |- { x e. dom I | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) }
85	30	eqeq1d 2238	. . . . . . . . . 10 |- ( ( ph /\ x e. dom I ) -> ( ( (iEdg ` U ) ` x ) = { N } <-> ( I ` x ) = { N } ) )
86	85	rabbidva 2788	. . . . . . . . 9 |- ( ph -> { x e. dom I | ( (iEdg ` U ) ` x ) = { N } } = { x e. dom I | ( I ` x ) = { N } } )
87	84, 86	eqtr3id 2276	. . . . . . . 8 |- ( ph -> { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x e. dom I | ( I ` x ) = { N } } )
88		df-rab 2517	. . . . . . . . 9 |- { x e. dom J | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) }
89	41	eqeq1d 2238	. . . . . . . . . 10 |- ( ( ph /\ x e. dom J ) -> ( ( (iEdg ` U ) ` x ) = { N } <-> ( J ` x ) = { N } ) )
90	89	rabbidva 2788	. . . . . . . . 9 |- ( ph -> { x e. dom J | ( (iEdg ` U ) ` x ) = { N } } = { x e. dom J | ( J ` x ) = { N } } )
91	88, 90	eqtr3id 2276	. . . . . . . 8 |- ( ph -> { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x e. dom J | ( J ` x ) = { N } } )
92	87, 91	uneq12d 3360	. . . . . . 7 |- ( ph -> ( { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } u. { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } ) = ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) )
93	80, 83, 92	3eqtrd 2266	. . . . . 6 |- ( ph -> { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } = ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) )
94	93	fveq2d 5639	. . . . 5 |- ( ph -> (♯ ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) = (♯ ` ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) ) )
95	48, 49, 50, 51, 52, 53, 54	vtxlpfi 16096	. . . . . 6 |- ( ph -> { x e. dom I | ( I ` x ) = { N } } e. Fin )
96	56, 57, 58, 59, 61, 62, 63	vtxlpfi 16096	. . . . . 6 |- ( ph -> { x e. dom J | ( J ` x ) = { N } } e. Fin )
97		ssrab2 3310	. . . . . . . . 9 |- { x e. dom I | ( I ` x ) = { N } } C_ dom I
98		ssrab2 3310	. . . . . . . . 9 |- { x e. dom J | ( J ` x ) = { N } } C_ dom J
99		ss2in 3433	. . . . . . . . 9 |- ( ( { x e. dom I | ( I ` x ) = { N } } C_ dom I /\ { x e. dom J | ( J ` x ) = { N } } C_ dom J ) -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ ( dom I i^i dom J ) )
100	97, 98, 99	mp2an 426	. . . . . . . 8 |- ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ ( dom I i^i dom J )
101	100, 26	sseqtrid 3275	. . . . . . 7 |- ( ph -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ (/) )
102		ss0 3533	. . . . . . 7 |- ( ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ (/) -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) = (/) )
103	101, 102	syl 14	. . . . . 6 |- ( ph -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) = (/) )
104		hashun 11058	. . . . . 6 |- ( ( { x e. dom I | ( I ` x ) = { N } } e. Fin /\ { x e. dom J | ( J ` x ) = { N } } e. Fin /\ ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) = (/) ) -> (♯ ` ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) ) = ( (♯ ` { x e. dom I | ( I ` x ) = { N } } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) )
105	95, 96, 103, 104	syl3anc 1271	. . . . 5 |- ( ph -> (♯ ` ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) ) = ( (♯ ` { x e. dom I | ( I ` x ) = { N } } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) )
106	94, 105	eqtrd 2262	. . . 4 |- ( ph -> (♯ ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) = ( (♯ ` { x e. dom I | ( I ` x ) = { N } } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) )
107	74, 106	oveq12d 6031	. . 3 |- ( ph -> ( (♯ ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) + (♯ ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) ) = ( ( (♯ ` { x e. dom I | N e. ( I ` x ) } ) + (♯ ` { x e. dom J | N e. ( J ` x ) } ) ) + ( (♯ ` { x e. dom I | ( I ` x ) = { N } } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) ) )
108		hashcl 11033	. . . . . 6 |- ( { x e. dom I | N e. ( I ` x ) } e. Fin -> (♯ ` { x e. dom I | N e. ( I ` x ) } ) e. NN0 )
109	55, 108	syl 14	. . . . 5 |- ( ph -> (♯ ` { x e. dom I | N e. ( I ` x ) } ) e. NN0 )
110	109	nn0cnd 9447	. . . 4 |- ( ph -> (♯ ` { x e. dom I | N e. ( I ` x ) } ) e. CC )
111		hashcl 11033	. . . . . 6 |- ( { x e. dom J | N e. ( J ` x ) } e. Fin -> (♯ ` { x e. dom J | N e. ( J ` x ) } ) e. NN0 )
112	64, 111	syl 14	. . . . 5 |- ( ph -> (♯ ` { x e. dom J | N e. ( J ` x ) } ) e. NN0 )
113	112	nn0cnd 9447	. . . 4 |- ( ph -> (♯ ` { x e. dom J | N e. ( J ` x ) } ) e. CC )
114		hashcl 11033	. . . . . 6 |- ( { x e. dom I | ( I ` x ) = { N } } e. Fin -> (♯ ` { x e. dom I | ( I ` x ) = { N } } ) e. NN0 )
115	95, 114	syl 14	. . . . 5 |- ( ph -> (♯ ` { x e. dom I | ( I ` x ) = { N } } ) e. NN0 )
116	115	nn0cnd 9447	. . . 4 |- ( ph -> (♯ ` { x e. dom I | ( I ` x ) = { N } } ) e. CC )
117		hashcl 11033	. . . . . 6 |- ( { x e. dom J | ( J ` x ) = { N } } e. Fin -> (♯ ` { x e. dom J | ( J ` x ) = { N } } ) e. NN0 )
118	96, 117	syl 14	. . . . 5 |- ( ph -> (♯ ` { x e. dom J | ( J ` x ) = { N } } ) e. NN0 )
119	118	nn0cnd 9447	. . . 4 |- ( ph -> (♯ ` { x e. dom J | ( J ` x ) = { N } } ) e. CC )
120	110, 113, 116, 119	add4d 8338	. . 3 |- ( ph -> ( ( (♯ ` { x e. dom I | N e. ( I ` x ) } ) + (♯ ` { x e. dom J | N e. ( J ` x ) } ) ) + ( (♯ ` { x e. dom I | ( I ` x ) = { N } } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) ) = ( ( (♯ ` { x e. dom I | N e. ( I ` x ) } ) + (♯ ` { x e. dom I | ( I ` x ) = { N } } ) ) + ( (♯ ` { x e. dom J | N e. ( J ` x ) } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) ) )
121	107, 120	eqtrd 2262	. 2 |- ( ph -> ( (♯ ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) + (♯ ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) ) = ( ( (♯ ` { x e. dom I | N e. ( I ` x ) } ) + (♯ ` { x e. dom I | ( I ` x ) = { N } } ) ) + ( (♯ ` { x e. dom J | N e. ( J ` x ) } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) ) )
122		eqid 2229	. . 3 |- (Vtx ` U ) = (Vtx ` U )
123		eqid 2229	. . 3 |- (iEdg ` U ) = (iEdg ` U )
124		eqid 2229	. . 3 |- dom (iEdg ` U ) = dom (iEdg ` U )
125		unfidisj 7107	. . . . 5 |- ( ( dom I e. Fin /\ dom J e. Fin /\ ( dom I i^i dom J ) = (/) ) -> ( dom I u. dom J ) e. Fin )
126	51, 59, 26, 125	syl3anc 1271	. . . 4 |- ( ph -> ( dom I u. dom J ) e. Fin )
127	5, 126	eqeltrd 2306	. . 3 |- ( ph -> dom (iEdg ` U ) e. Fin )
128		vtxdun.vu	. . . 4 |- ( ph -> (Vtx ` U ) = V )
129	128, 52	eqeltrd 2306	. . 3 |- ( ph -> (Vtx ` U ) e. Fin )
130	53, 128	eleqtrrd 2309	. . 3 |- ( ph -> N e. (Vtx ` U ) )
131	122	1vgrex 15861	. . . . 5 |- ( N e. (Vtx ` U ) -> U e. _V )
132	130, 131	syl 14	. . . 4 |- ( ph -> U e. _V )
133	54, 63, 49, 57, 48, 60, 26, 132, 128, 2	upgrun 15965	. . 3 |- ( ph -> U e. UPGraph )
134	122, 123, 124, 127, 129, 130, 133	vtxdgfifival 16097	. 2 |- ( ph -> ( (VtxDeg ` U ) ` N ) = ( (♯ ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) + (♯ ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) ) )
135	48, 49, 50, 51, 52, 53, 54	vtxdgfifival 16097	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` N ) = ( (♯ ` { x e. dom I | N e. ( I ` x ) } ) + (♯ ` { x e. dom I | ( I ` x ) = { N } } ) ) )
136	56, 57, 58, 59, 61, 62, 63	vtxdgfifival 16097	. . 3 |- ( ph -> ( (VtxDeg ` H ) ` N ) = ( (♯ ` { x e. dom J | N e. ( J ` x ) } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) )
137	135, 136	oveq12d 6031	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` N ) + ( (VtxDeg ` H ) ` N ) ) = ( ( (♯ ` { x e. dom I | N e. ( I ` x ) } ) + (♯ ` { x e. dom I | ( I ` x ) = { N } } ) ) + ( (♯ ` { x e. dom J | N e. ( J ` x ) } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) ) )
138	121, 134, 137	3eqtr4d 2272	1 |- ( ph -> ( (VtxDeg ` U ) ` N ) = ( ( (VtxDeg ` G ) ` N ) + ( (VtxDeg ` H ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   _Vcvv 2800   u. cun 3196   i^i cin 3197   C_ wss 3198   (/)c0 3492   {csn 3667   dom cdm 4723   Fun wfun 5318   Fn wfn 5319   ` cfv 5324  (class class class)co 6013   Fincfn 6904   + caddc 8025   NN0cn0 9392  ♯chash 11027  Vtxcvtx 15853  iEdgciedg 15854  UPGraphcupgr 15932  VtxDegcvtxdg 16092

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-2o 6578  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-z 9470  df-dec 9602  df-uz 9746  df-xadd 9998  df-ihash 11028  df-ndx 13075  df-slot 13076  df-base 13078  df-edgf 15846  df-vtx 15855  df-iedg 15856  df-upgren 15934  df-vtxdg 16093

This theorem is referenced by: (None)