Theorem vtxdfifiun 16743

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 4985	. . . . . . . . . . . . . 14 |- ( ph -> dom (iEdg ` U ) = dom ( I u. J ) )
4		dmun 4990	. . . . . . . . . . . . . 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 5700	. . . . . . . . . . . . 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 5411	. . . . . . . . . . . . . 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 5411	. . . . . . . . . . . . . 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 5772	. . . . . . . . . . . . 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 5773	. . . . . . . . . . . . 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 5702	. . . . 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 16735	. . . . . 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 16735	. . . . . 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 11267	. . . . . 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 5702	. . . . 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 16736	. . . . . 6 |- ( ph -> { x e. dom I | ( I ` x ) = { N } } e. Fin )
96	56, 57, 58, 59, 61, 62, 63	vtxlpfi 16736	. . . . . 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 11267	. . . . . 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 6106	. . 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 11242	. . . . . 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 9633	. . . 4 |- ( ph -> (♯ ` { x e. dom I | N e. ( I ` x ) } ) e. CC )
111		hashcl 11242	. . . . . 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 9633	. . . 4 |- ( ph -> (♯ ` { x e. dom J | N e. ( J ` x ) } ) e. CC )
114		hashcl 11242	. . . . . 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 9633	. . . 4 |- ( ph -> (♯ ` { x e. dom I | ( I ` x ) = { N } } ) e. CC )
117		hashcl 11242	. . . . . 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 9633	. . . 4 |- ( ph -> (♯ ` { x e. dom J | ( J ` x ) = { N } } ) e. CC )
120	110, 113, 116, 119	add4d 8502	. . 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 7233	. . . . 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 16466	. . . . 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 16572	. . 3 |- ( ph -> U e. UPGraph )
134	122, 123, 124, 127, 129, 130, 133	vtxdgfifival 16737	. 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 16737	. . 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 16737	. . 3 |- ( ph -> ( (VtxDeg ` H ) ` N ) = ( (♯ ` { x e. dom J | N e. ( J ` x ) } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) )
137	135, 136	oveq12d 6106	. 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 ) ) )

This proof depends on syntax axioms:   -> 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 3709   dom cdm 4776   Fun wfun 5374   Fn wfn 5375   ` cfv 5380  (class class class)co 6088   Fincfn 7026   + caddc 8188   NN0cn0 9574  ♯chash 11236  Vtxcvtx 16458  iEdgciedg 16459  UPGraphcupgr 16537  VtxDegcvtxdg 16732

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-dec 9789  df-uz 9938  df-xadd 10192  df-ihash 11237  df-ndx 13413  df-slot 13414  df-base 13416  df-edgf 16451  df-vtx 16460  df-iedg 16461  df-upgren 16539  df-vtxdg 16733

This theorem is used by:  p1evtxdeqfilem  16757  trlsegvdegfi  16913