Theorem nbgraopALT 24642

Description: Alternate proof of nbgraop 24641 using mpt2xopoveq 6965, but being longer. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nbgraopALT	|- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( <. V , E >. Neighbors N ) = { n e. V | { N , n } e. ran E } )

Distinct variable groups:   n, V   n, E   n, N   n, Y   n, Z

Proof of Theorem nbgraopALT

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nbgra 24638	. . 3 |- Neighbors = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | { y , n } e. ran ( 2nd ` x ) } )
2	1	mpt2xopoveq 6965	. 2 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( <. V , E >. Neighbors N ) = { n e. V | [. <. V , E >. / x ]. [. N / y ]. { y , n } e. ran ( 2nd ` x ) } )
3		sbcel1g 3837	. . . . . 6 |- ( N e. V -> ( [. N / y ]. { y , n } e. ran ( 2nd ` x ) <-> [_ N / y ]_ { y , n } e. ran ( 2nd ` x ) ) )
4	3	adantl 466	. . . . 5 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( [. N / y ]. { y , n } e. ran ( 2nd ` x ) <-> [_ N / y ]_ { y , n } e. ran ( 2nd ` x ) ) )
5	4	sbcbidv 3386	. . . 4 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( [. <. V , E >. / x ]. [. N / y ]. { y , n } e. ran ( 2nd ` x ) <-> [. <. V , E >. / x ]. [_ N / y ]_ { y , n } e. ran ( 2nd ` x ) ) )
6		sbcel2 3839	. . . . 5 |- ( [. <. V , E >. / x ]. [_ N / y ]_ { y , n } e. ran ( 2nd ` x ) <-> [_ N / y ]_ { y , n } e. [_ <. V , E >. / x ]_ ran ( 2nd ` x ) )
7	6	a1i 11	. . . 4 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( [. <. V , E >. / x ]. [_ N / y ]_ { y , n } e. ran ( 2nd ` x ) <-> [_ N / y ]_ { y , n } e. [_ <. V , E >. / x ]_ ran ( 2nd ` x ) ) )
8		df-pr 4035	. . . . . . . 8 |- { y , n } = ( { y } u. { n } )
9	8	a1i 11	. . . . . . 7 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> { y , n } = ( { y } u. { n } ) )
10	9	csbeq2dv 3843	. . . . . 6 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> [_ N / y ]_ { y , n } = [_ N / y ]_ ( { y } u. { n } ) )
11		nfcv 2619	. . . . . . . . 9 |- F/_ y ( { N } u. { n } )
12	11	a1i 11	. . . . . . . 8 |- ( N e. V -> F/_ y ( { N } u. { n } ) )
13		sneq 4042	. . . . . . . . 9 |- ( y = N -> { y } = { N } )
14	13	uneq1d 3653	. . . . . . . 8 |- ( y = N -> ( { y } u. { n } ) = ( { N } u. { n } ) )
15	12, 14	csbiegf 3454	. . . . . . 7 |- ( N e. V -> [_ N / y ]_ ( { y } u. { n } ) = ( { N } u. { n } ) )
16	15	adantl 466	. . . . . 6 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> [_ N / y ]_ ( { y } u. { n } ) = ( { N } u. { n } ) )
17		df-pr 4035	. . . . . . . 8 |- { N , n } = ( { N } u. { n } )
18	17	eqcomi 2470	. . . . . . 7 |- ( { N } u. { n } ) = { N , n }
19	18	a1i 11	. . . . . 6 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( { N } u. { n } ) = { N , n } )
20	10, 16, 19	3eqtrd 2502	. . . . 5 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> [_ N / y ]_ { y , n } = { N , n } )
21		csbrn 5474	. . . . . . 7 |- [_ <. V , E >. / x ]_ ran ( 2nd ` x ) = ran [_ <. V , E >. / x ]_ ( 2nd ` x )
22	21	a1i 11	. . . . . 6 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> [_ <. V , E >. / x ]_ ran ( 2nd ` x ) = ran [_ <. V , E >. / x ]_ ( 2nd ` x ) )
23		opex 4720	. . . . . . . . 9 |- <. V , E >. e. _V
24		csbfv2g 5908	. . . . . . . . 9 |- ( <. V , E >. e. _V -> [_ <. V , E >. / x ]_ ( 2nd ` x ) = ( 2nd ` [_ <. V , E >. / x ]_ x ) )
25	23, 24	mp1i 12	. . . . . . . 8 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> [_ <. V , E >. / x ]_ ( 2nd ` x ) = ( 2nd ` [_ <. V , E >. / x ]_ x ) )
26		csbvarg 3855	. . . . . . . . . 10 |- ( <. V , E >. e. _V -> [_ <. V , E >. / x ]_ x = <. V , E >. )
27	23, 26	mp1i 12	. . . . . . . . 9 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> [_ <. V , E >. / x ]_ x = <. V , E >. )
28	27	fveq2d 5876	. . . . . . . 8 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( 2nd ` [_ <. V , E >. / x ]_ x ) = ( 2nd ` <. V , E >. ) )
29		op2ndg 6812	. . . . . . . . 9 |- ( ( V e. Y /\ E e. Z ) -> ( 2nd ` <. V , E >. ) = E )
30	29	adantr 465	. . . . . . . 8 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( 2nd ` <. V , E >. ) = E )
31	25, 28, 30	3eqtrd 2502	. . . . . . 7 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> [_ <. V , E >. / x ]_ ( 2nd ` x ) = E )
32	31	rneqd 5240	. . . . . 6 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ran [_ <. V , E >. / x ]_ ( 2nd ` x ) = ran E )
33	22, 32	eqtrd 2498	. . . . 5 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> [_ <. V , E >. / x ]_ ran ( 2nd ` x ) = ran E )
34	20, 33	eleq12d 2539	. . . 4 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( [_ N / y ]_ { y , n } e. [_ <. V , E >. / x ]_ ran ( 2nd ` x ) <-> { N , n } e. ran E ) )
35	5, 7, 34	3bitrd 279	. . 3 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( [. <. V , E >. / x ]. [. N / y ]. { y , n } e. ran ( 2nd ` x ) <-> { N , n } e. ran E ) )
36	35	rabbidv 3101	. 2 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> { n e. V | [. <. V , E >. / x ]. [. N / y ]. { y , n } e. ran ( 2nd ` x ) } = { n e. V | { N , n } e. ran E } )
37	2, 36	eqtrd 2498	1 |- ( ( ( V e. Y /\ E e. Z ) /\ N e. V ) -> ( <. V , E >. Neighbors N ) = { n e. V | { N , n } e. ran E } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   F/_wnfc 2605   {crab 2811   _Vcvv 3109   [.wsbc 3327   [_csb 3430   u. cun 3469   {csn 4032   {cpr 4034   <.cop 4038   ran crn 5009   ` cfv 5594  (class class class)co 6296   2ndc2nd 6798   Neighbors cnbgra 24635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-nbgra 24638

This theorem is referenced by: (None)