Theorem vtxdfifiun 16538

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 4983	. . . . . . . . . . . . . 14 |- ( ph -> dom (iEdg ` U ) = dom ( I u. J ) )
4		dmun 4988	. . . . . . . . . . . . . 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 5697	. . . . . . . . . . . . 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 5408	. . . . . . . . . . . . . 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 5408	. . . . . . . . . . . . . 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 5769	. . . . . . . . . . . . 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 5770	. . . . . . . . . . . . 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 5699	. . . . 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 16530	. . . . . 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 16530	. . . . . 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 11245	. . . . . 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 5699	. . . . 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 16531	. . . . . 6 |- ( ph -> { x e. dom I | ( I ` x ) = { N } } e. Fin )
96	56, 57, 58, 59, 61, 62, 63	vtxlpfi 16531	. . . . . 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 11245	. . . . . 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 6103	. . 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 11220	. . . . . 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 9622	. . . 4 |- ( ph -> (♯ ` { x e. dom I | N e. ( I ` x ) } ) e. CC )
111		hashcl 11220	. . . . . 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 9622	. . . 4 |- ( ph -> (♯ ` { x e. dom J | N e. ( J ` x ) } ) e. CC )
114		hashcl 11220	. . . . . 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 9622	. . . 4 |- ( ph -> (♯ ` { x e. dom I | ( I ` x ) = { N } } ) e. CC )
117		hashcl 11220	. . . . . 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 9622	. . . 4 |- ( ph -> (♯ ` { x e. dom J | ( J ` x ) = { N } } ) e. CC )
120	110, 113, 116, 119	add4d 8495	. . 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 7229	. . . . 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 16261	. . . . 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 16367	. . 3 |- ( ph -> U e. UPGraph )
134	122, 123, 124, 127, 129, 130, 133	vtxdgfifival 16532	. 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 16532	. . 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 16532	. . 3 |- ( ph -> ( (VtxDeg ` H ) ` N ) = ( (♯ ` { x e. dom J | N e. ( J ` x ) } ) + (♯ ` { x e. dom J | ( J ` x ) = { N } } ) ) )
137	135, 136	oveq12d 6103	. 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 4774   Fun wfun 5371   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Fincfn 7022   + caddc 8182   NN0cn0 9563  ♯chash 11214  Vtxcvtx 16253  iEdgciedg 16254  UPGraphcupgr 16332  VtxDegcvtxdg 16527

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-xadd 10175  df-ihash 11215  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-upgren 16334  df-vtxdg 16528

This theorem is used by:  p1evtxdeqfilem  16552  trlsegvdegfi  16708