Theorem vtxdfifiun 16452

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 2537	. . . . . . . 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 4978	. . . . . . . . . . . . . 14 |- ( ph -> dom (iEdg ` U ) = dom ( I u. J ) )
4		dmun 4983	. . . . . . . . . . . . . 14 |- dom ( I u. J ) = ( dom I u. dom J )
5	3, 4	eqtrdi 2287	. . . . . . . . . . . . 13 |- ( ph -> dom (iEdg ` U ) = ( dom I u. dom J ) )
6	5	eleq2d 2308	. . . . . . . . . . . 12 |- ( ph -> ( x e. dom (iEdg ` U ) <-> x e. ( dom I u. dom J ) ) )
7		elun 3370	. . . . . . . . . . . 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 469	. . . . . . . . . 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 831	. . . . . . . . . 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 2358	. . . . . . . 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 2283	. . . . . . 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 3498	. . . . . . . . 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 2242	. . . . . . . 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 2537	. . . . . . . . 9 |- { x e. dom I | N e. ( (iEdg ` U ) ` x ) } = { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) }
18	2	fveq1d 5692	. . . . . . . . . . . . 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 5403	. . . . . . . . . . . . . 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 5403	. . . . . . . . . . . . . 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 5763	. . . . . . . . . . . . 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 1278	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom I ) -> ( ( I u. J ) ` x ) = ( I ` x ) )
30	19, 29	eqtrd 2271	. . . . . . . . . . 11 |- ( ( ph /\ x e. dom I ) -> ( (iEdg ` U ) ` x ) = ( I ` x ) )
31	30	eleq2d 2308	. . . . . . . . . 10 |- ( ( ph /\ x e. dom I ) -> ( N e. ( (iEdg ` U ) ` x ) <-> N e. ( I ` x ) ) )
32	31	rabbidva 2809	. . . . . . . . 9 |- ( ph -> { x e. dom I | N e. ( (iEdg ` U ) ` x ) } = { x e. dom I | N e. ( I ` x ) } )
33	17, 32	eqtr3id 2285	. . . . . . . 8 |- ( ph -> { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } = { x e. dom I | N e. ( I ` x ) } )
34		df-rab 2537	. . . . . . . . 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 5764	. . . . . . . . . . . . 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 1278	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom J ) -> ( ( I u. J ) ` x ) = ( J ` x ) )
41	35, 40	eqtrd 2271	. . . . . . . . . . 11 |- ( ( ph /\ x e. dom J ) -> ( (iEdg ` U ) ` x ) = ( J ` x ) )
42	41	eleq2d 2308	. . . . . . . . . 10 |- ( ( ph /\ x e. dom J ) -> ( N e. ( (iEdg ` U ) ` x ) <-> N e. ( J ` x ) ) )
43	42	rabbidva 2809	. . . . . . . . 9 |- ( ph -> { x e. dom J | N e. ( (iEdg ` U ) ` x ) } = { x e. dom J | N e. ( J ` x ) } )
44	34, 43	eqtr3id 2285	. . . . . . . 8 |- ( ph -> { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } = { x e. dom J | N e. ( J ` x ) } )
45	33, 44	uneq12d 3384	. . . . . . 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 2275	. . . . . 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 5694	. . . . 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 2238	. . . . . . 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 16444	. . . . . 6 |- ( ph -> { x e. dom I | N e. ( I ` x ) } e. Fin )
56		eqid 2238	. . . . . . 7 |- (Vtx ` H ) = (Vtx ` H )
57		vtxdun.j	. . . . . . 7 |- J = (iEdg ` H )
58		eqid 2238	. . . . . . 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 2315	. . . . . . 7 |- ( ph -> (Vtx ` H ) e. Fin )
62	53, 60	eleqtrrd 2318	. . . . . . 7 |- ( ph -> N e. (Vtx ` H ) )
63		vtxdfifiun.h	. . . . . . 7 |- ( ph -> H e. UPGraph )
64	56, 57, 58, 59, 61, 62, 63	vtxedgfi 16444	. . . . . 6 |- ( ph -> { x e. dom J | N e. ( J ` x ) } e. Fin )
65		ssrab2 3333	. . . . . . . . 9 |- { x e. dom I | N e. ( I ` x ) } C_ dom I
66		ssrab2 3333	. . . . . . . . 9 |- { x e. dom J | N e. ( J ` x ) } C_ dom J
67		ss2in 3459	. . . . . . . . 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 430	. . . . . . . 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 3298	. . . . . . 7 |- ( ph -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ (/) )
70		ss0 3563	. . . . . . 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 11223	. . . . . 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 1278	. . . . 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 2271	. . . 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 2537	. . . . . . . 8 |- { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom (iEdg ` U ) /\ ( (iEdg ` U ) ` x ) = { N } ) }
76	8	anbi1d 469	. . . . . . . . . 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 831	. . . . . . . . . 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 2358	. . . . . . . 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 2283	. . . . . . 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 3498	. . . . . . . . 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 2242	. . . . . . . 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 2537	. . . . . . . . 9 |- { x e. dom I | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) }
85	30	eqeq1d 2247	. . . . . . . . . 10 |- ( ( ph /\ x e. dom I ) -> ( ( (iEdg ` U ) ` x ) = { N } <-> ( I ` x ) = { N } ) )
86	85	rabbidva 2809	. . . . . . . . 9 |- ( ph -> { x e. dom I | ( (iEdg ` U ) ` x ) = { N } } = { x e. dom I | ( I ` x ) = { N } } )
87	84, 86	eqtr3id 2285	. . . . . . . 8 |- ( ph -> { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x e. dom I | ( I ` x ) = { N } } )
88		df-rab 2537	. . . . . . . . 9 |- { x e. dom J | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) }
89	41	eqeq1d 2247	. . . . . . . . . 10 |- ( ( ph /\ x e. dom J ) -> ( ( (iEdg ` U ) ` x ) = { N } <-> ( J ` x ) = { N } ) )
90	89	rabbidva 2809	. . . . . . . . 9 |- ( ph -> { x e. dom J | ( (iEdg ` U ) ` x ) = { N } } = { x e. dom J | ( J ` x ) = { N } } )
91	88, 90	eqtr3id 2285	. . . . . . . 8 |- ( ph -> { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x e. dom J | ( J ` x ) = { N } } )
92	87, 91	uneq12d 3384	. . . . . . 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 2275	. . . . . 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 5694	. . . . 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 16445	. . . . . 6 |- ( ph -> { x e. dom I | ( I ` x ) = { N } } e. Fin )
96	56, 57, 58, 59, 61, 62, 63	vtxlpfi 16445	. . . . . 6 |- ( ph -> { x e. dom J | ( J ` x ) = { N } } e. Fin )
97		ssrab2 3333	. . . . . . . . 9 |- { x e. dom I | ( I ` x ) = { N } } C_ dom I
98		ssrab2 3333	. . . . . . . . 9 |- { x e. dom J | ( J ` x ) = { N } } C_ dom J
99		ss2in 3459	. . . . . . . . 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 430	. . . . . . . 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 3298	. . . . . . 7 |- ( ph -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ (/) )
102		ss0 3563	. . . . . . 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 11223	. . . . . 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 1278	. . . . 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 2271	. . . 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 6093	. . 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 11198	. . . . . 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 9601	. . . 4 |- ( ph -> (♯ ` { x e. dom I | N e. ( I ` x ) } ) e. CC )
111		hashcl 11198	. . . . . 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 9601	. . . 4 |- ( ph -> (♯ ` { x e. dom J | N e. ( J ` x ) } ) e. CC )
114		hashcl 11198	. . . . . 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 9601	. . . 4 |- ( ph -> (♯ ` { x e. dom I | ( I ` x ) = { N } } ) e. CC )
117		hashcl 11198	. . . . . 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 9601	. . . 4 |- ( ph -> (♯ ` { x e. dom J | ( J ` x ) = { N } } ) e. CC )
120	110, 113, 116, 119	add4d 8485	. . 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 2271	. 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 2238	. . 3 |- (Vtx ` U ) = (Vtx ` U )
123		eqid 2238	. . 3 |- (iEdg ` U ) = (iEdg ` U )
124		eqid 2238	. . 3 |- dom (iEdg ` U ) = dom (iEdg ` U )
125		unfidisj 7219	. . . . 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 1278	. . . 4 |- ( ph -> ( dom I u. dom J ) e. Fin )
127	5, 126	eqeltrd 2315	. . 3 |- ( ph -> dom (iEdg ` U ) e. Fin )
128		vtxdun.vu	. . . 4 |- ( ph -> (Vtx ` U ) = V )
129	128, 52	eqeltrd 2315	. . 3 |- ( ph -> (Vtx ` U ) e. Fin )
130	53, 128	eleqtrrd 2318	. . 3 |- ( ph -> N e. (Vtx ` U ) )
131	122	1vgrex 16175	. . . . 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 16281	. . 3 |- ( ph -> U e. UPGraph )
134	122, 123, 124, 127, 129, 130, 133	vtxdgfifival 16446	. 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 16446	. . 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 16446	. . 3 |- ( ph -> ( (VtxDeg ` H ) ` N ) = ( (♯ ` { x e. dom J | N e. ( J ` x ) } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) )
137	135, 136	oveq12d 6093	. 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 2281	1 |- ( ph -> ( (VtxDeg ` U ) ` N ) = ( ( (VtxDeg ` G ) ` N ) + ( (VtxDeg ` H ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   _Vcvv 2821   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   {csn 3705   dom cdm 4769   Fun wfun 5366   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Fincfn 7012   + caddc 8172   NN0cn0 9542  ♯chash 11192  Vtxcvtx 16167  iEdgciedg 16168  UPGraphcupgr 16246  VtxDegcvtxdg 16441

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-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  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-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  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-nel 2516  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-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-2o 6678  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  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-z 9624  df-dec 9757  df-uz 9901  df-xadd 10154  df-ihash 11193  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-upgren 16248  df-vtxdg 16442

This theorem is referenced by:  p1evtxdeqfilem  16466  trlsegvdegfi  16622