Theorem vtxdfifiun 16147

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 2519	. . . . . . . 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 4933	. . . . . . . . . . . . . 14 |- ( ph -> dom (iEdg ` U ) = dom ( I u. J ) )
4		dmun 4938	. . . . . . . . . . . . . 14 |- dom ( I u. J ) = ( dom I u. dom J )
5	3, 4	eqtrdi 2280	. . . . . . . . . . . . 13 |- ( ph -> dom (iEdg ` U ) = ( dom I u. dom J ) )
6	5	eleq2d 2301	. . . . . . . . . . . 12 |- ( ph -> ( x e. dom (iEdg ` U ) <-> x e. ( dom I u. dom J ) ) )
7		elun 3348	. . . . . . . . . . . 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 826	. . . . . . . . . 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 2349	. . . . . . . 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 2276	. . . . . . 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 3474	. . . . . . . . 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 2235	. . . . . . . 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 2519	. . . . . . . . 9 |- { x e. dom I | N e. ( (iEdg ` U ) ` x ) } = { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) }
18	2	fveq1d 5641	. . . . . . . . . . . . 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 5357	. . . . . . . . . . . . . 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 5357	. . . . . . . . . . . . . 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 5712	. . . . . . . . . . . . 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 1273	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom I ) -> ( ( I u. J ) ` x ) = ( I ` x ) )
30	19, 29	eqtrd 2264	. . . . . . . . . . 11 |- ( ( ph /\ x e. dom I ) -> ( (iEdg ` U ) ` x ) = ( I ` x ) )
31	30	eleq2d 2301	. . . . . . . . . 10 |- ( ( ph /\ x e. dom I ) -> ( N e. ( (iEdg ` U ) ` x ) <-> N e. ( I ` x ) ) )
32	31	rabbidva 2790	. . . . . . . . 9 |- ( ph -> { x e. dom I | N e. ( (iEdg ` U ) ` x ) } = { x e. dom I | N e. ( I ` x ) } )
33	17, 32	eqtr3id 2278	. . . . . . . 8 |- ( ph -> { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } = { x e. dom I | N e. ( I ` x ) } )
34		df-rab 2519	. . . . . . . . 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 5713	. . . . . . . . . . . . 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 1273	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom J ) -> ( ( I u. J ) ` x ) = ( J ` x ) )
41	35, 40	eqtrd 2264	. . . . . . . . . . 11 |- ( ( ph /\ x e. dom J ) -> ( (iEdg ` U ) ` x ) = ( J ` x ) )
42	41	eleq2d 2301	. . . . . . . . . 10 |- ( ( ph /\ x e. dom J ) -> ( N e. ( (iEdg ` U ) ` x ) <-> N e. ( J ` x ) ) )
43	42	rabbidva 2790	. . . . . . . . 9 |- ( ph -> { x e. dom J | N e. ( (iEdg ` U ) ` x ) } = { x e. dom J | N e. ( J ` x ) } )
44	34, 43	eqtr3id 2278	. . . . . . . 8 |- ( ph -> { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } = { x e. dom J | N e. ( J ` x ) } )
45	33, 44	uneq12d 3362	. . . . . . 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 2268	. . . . . 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 5643	. . . . 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 2231	. . . . . . 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 16139	. . . . . 6 |- ( ph -> { x e. dom I | N e. ( I ` x ) } e. Fin )
56		eqid 2231	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
57		vtxdun.j	. . . . . . 7 |- J = (iEdg ` H )
58		eqid 2231	. . . . . . 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 2308	. . . . . . 7 |- ( ph -> (Vtx ` H ) e. Fin )
62	53, 60	eleqtrrd 2311	. . . . . . 7 |- ( ph -> N e. (Vtx ` H ) )
63		vtxdfifiun.h	. . . . . . 7 |- ( ph -> H e. UPGraph )
64	56, 57, 58, 59, 61, 62, 63	vtxedgfi 16139	. . . . . 6 |- ( ph -> { x e. dom J | N e. ( J ` x ) } e. Fin )
65		ssrab2 3312	. . . . . . . . 9 |- { x e. dom I | N e. ( I ` x ) } C_ dom I
66		ssrab2 3312	. . . . . . . . 9 |- { x e. dom J | N e. ( J ` x ) } C_ dom J
67		ss2in 3435	. . . . . . . . 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 3277	. . . . . . 7 |- ( ph -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ (/) )
70		ss0 3535	. . . . . . 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 11067	. . . . . 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 1273	. . . . 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 2264	. . . 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 2519	. . . . . . . 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 826	. . . . . . . . . 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 2349	. . . . . . . 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 2276	. . . . . . 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 3474	. . . . . . . . 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 2235	. . . . . . . 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 2519	. . . . . . . . 9 |- { x e. dom I | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) }
85	30	eqeq1d 2240	. . . . . . . . . 10 |- ( ( ph /\ x e. dom I ) -> ( ( (iEdg ` U ) ` x ) = { N } <-> ( I ` x ) = { N } ) )
86	85	rabbidva 2790	. . . . . . . . 9 |- ( ph -> { x e. dom I | ( (iEdg ` U ) ` x ) = { N } } = { x e. dom I | ( I ` x ) = { N } } )
87	84, 86	eqtr3id 2278	. . . . . . . 8 |- ( ph -> { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x e. dom I | ( I ` x ) = { N } } )
88		df-rab 2519	. . . . . . . . 9 |- { x e. dom J | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) }
89	41	eqeq1d 2240	. . . . . . . . . 10 |- ( ( ph /\ x e. dom J ) -> ( ( (iEdg ` U ) ` x ) = { N } <-> ( J ` x ) = { N } ) )
90	89	rabbidva 2790	. . . . . . . . 9 |- ( ph -> { x e. dom J | ( (iEdg ` U ) ` x ) = { N } } = { x e. dom J | ( J ` x ) = { N } } )
91	88, 90	eqtr3id 2278	. . . . . . . 8 |- ( ph -> { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x e. dom J | ( J ` x ) = { N } } )
92	87, 91	uneq12d 3362	. . . . . . 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 2268	. . . . . 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 5643	. . . . 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 16140	. . . . . 6 |- ( ph -> { x e. dom I | ( I ` x ) = { N } } e. Fin )
96	56, 57, 58, 59, 61, 62, 63	vtxlpfi 16140	. . . . . 6 |- ( ph -> { x e. dom J | ( J ` x ) = { N } } e. Fin )
97		ssrab2 3312	. . . . . . . . 9 |- { x e. dom I | ( I ` x ) = { N } } C_ dom I
98		ssrab2 3312	. . . . . . . . 9 |- { x e. dom J | ( J ` x ) = { N } } C_ dom J
99		ss2in 3435	. . . . . . . . 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 3277	. . . . . . 7 |- ( ph -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ (/) )
102		ss0 3535	. . . . . . 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 11067	. . . . . 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 1273	. . . . 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 2264	. . . 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 6035	. . 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 11042	. . . . . 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 9456	. . . 4 |- ( ph -> (♯ ` { x e. dom I | N e. ( I ` x ) } ) e. CC )
111		hashcl 11042	. . . . . 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 9456	. . . 4 |- ( ph -> (♯ ` { x e. dom J | N e. ( J ` x ) } ) e. CC )
114		hashcl 11042	. . . . . 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 9456	. . . 4 |- ( ph -> (♯ ` { x e. dom I | ( I ` x ) = { N } } ) e. CC )
117		hashcl 11042	. . . . . 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 9456	. . . 4 |- ( ph -> (♯ ` { x e. dom J | ( J ` x ) = { N } } ) e. CC )
120	110, 113, 116, 119	add4d 8347	. . 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 2264	. 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 2231	. . 3 |- (Vtx ` U ) = (Vtx ` U )
123		eqid 2231	. . 3 |- (iEdg ` U ) = (iEdg ` U )
124		eqid 2231	. . 3 |- dom (iEdg ` U ) = dom (iEdg ` U )
125		unfidisj 7113	. . . . 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 1273	. . . 4 |- ( ph -> ( dom I u. dom J ) e. Fin )
127	5, 126	eqeltrd 2308	. . 3 |- ( ph -> dom (iEdg ` U ) e. Fin )
128		vtxdun.vu	. . . 4 |- ( ph -> (Vtx ` U ) = V )
129	128, 52	eqeltrd 2308	. . 3 |- ( ph -> (Vtx ` U ) e. Fin )
130	53, 128	eleqtrrd 2311	. . 3 |- ( ph -> N e. (Vtx ` U ) )
131	122	1vgrex 15870	. . . . 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 15976	. . 3 |- ( ph -> U e. UPGraph )
134	122, 123, 124, 127, 129, 130, 133	vtxdgfifival 16141	. 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 16141	. . 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 16141	. . 3 |- ( ph -> ( (VtxDeg ` H ) ` N ) = ( (♯ ` { x e. dom J | N e. ( J ` x ) } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) )
137	135, 136	oveq12d 6035	. 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 2274	1 |- ( ph -> ( (VtxDeg ` U ) ` N ) = ( ( (VtxDeg ` G ) ` N ) + ( (VtxDeg ` H ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   _Vcvv 2802   u. cun 3198   i^i cin 3199   C_ wss 3200   (/)c0 3494   {csn 3669   dom cdm 4725   Fun wfun 5320   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   Fincfn 6908   + caddc 8034   NN0cn0 9401  ♯chash 11036  Vtxcvtx 15862  iEdgciedg 15863  UPGraphcupgr 15941  VtxDegcvtxdg 16136

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-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  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-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  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-nel 2498  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-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-2o 6582  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  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-z 9479  df-dec 9611  df-uz 9755  df-xadd 10007  df-ihash 11037  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-upgren 15943  df-vtxdg 16137

This theorem is referenced by:  p1evtxdeqfilem  16161