Theorem nbgraf1olem3 23505

Description: Lemma 3 for nbgraf1o 23509. The restricted iota of an edge is the function value of the converse applied to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.)

Hypotheses

Ref	Expression
nbgraf1o.n	|- N = ( <. V , E >. Neighbors U )
nbgraf1o.i	|- I = { i e. dom E | U e. ( E ` i ) }
nbgraf1o.f	|- F = ( n e. N |-> ( iota_ i e. I ( E ` i ) = { U , n } ) )

Assertion

Ref	Expression
nbgraf1olem3	|- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( iota_ i e. I ( E ` i ) = { U , M } ) = ( `' E ` { U , M } ) )

Distinct variable groups:   i, E, n   U, i, n   i, V, n   i, I, n   n, N   i, M

Allowed substitution hints:   F( i, n)   M( n)   N( i)

Proof of Theorem nbgraf1olem3

Step	Hyp	Ref	Expression
1		nbgraf1o.n	. . . . 5 |- N = ( <. V , E >. Neighbors U )
2	1	eleq2i 2532	. . . 4 |- ( M e. N <-> M e. ( <. V , E >. Neighbors U ) )
3		nbgracnvfv 23502	. . . 4 |- ( ( V USGrph E /\ M e. ( <. V , E >. Neighbors U ) ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
4	2, 3	sylan2b 475	. . 3 |- ( ( V USGrph E /\ M e. N ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
5	4	3adant2 1007	. 2 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
6		usgraf1o 23434	. . . . . . 7 |- ( V USGrph E -> E : dom E -1-1-onto-> ran E )
7	6	3ad2ant1 1009	. . . . . 6 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> E : dom E -1-1-onto-> ran E )
8		nbgraeledg 23494	. . . . . . . . . . . 12 |- ( V USGrph E -> ( M e. ( <. V , E >. Neighbors U ) <-> { M , U } e. ran E ) )
9		prcom 4062	. . . . . . . . . . . . 13 |- { M , U } = { U , M }
10	9	eleq1i 2531	. . . . . . . . . . . 12 |- ( { M , U } e. ran E <-> { U , M } e. ran E )
11	8, 10	syl6bb 261	. . . . . . . . . . 11 |- ( V USGrph E -> ( M e. ( <. V , E >. Neighbors U ) <-> { U , M } e. ran E ) )
12	11	biimpcd 224	. . . . . . . . . 10 |- ( M e. ( <. V , E >. Neighbors U ) -> ( V USGrph E -> { U , M } e. ran E ) )
13	12	a1d 25	. . . . . . . . 9 |- ( M e. ( <. V , E >. Neighbors U ) -> ( U e. V -> ( V USGrph E -> { U , M } e. ran E ) ) )
14	13, 1	eleq2s 2562	. . . . . . . 8 |- ( M e. N -> ( U e. V -> ( V USGrph E -> { U , M } e. ran E ) ) )
15	14	com13 80	. . . . . . 7 |- ( V USGrph E -> ( U e. V -> ( M e. N -> { U , M } e. ran E ) ) )
16	15	3imp 1182	. . . . . 6 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> { U , M } e. ran E )
17		f1ocnvdm 6099	. . . . . 6 |- ( ( E : dom E -1-1-onto-> ran E /\ { U , M } e. ran E ) -> ( `' E ` { U , M } ) e. dom E )
18	7, 16, 17	syl2anc 661	. . . . 5 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( `' E ` { U , M } ) e. dom E )
19		prid1g 4090	. . . . . . . 8 |- ( U e. V -> U e. { U , M } )
20		eleq2 2527	. . . . . . . 8 |- ( ( E ` ( `' E ` { U , M } ) ) = { U , M } -> ( U e. ( E ` ( `' E ` { U , M } ) ) <-> U e. { U , M } ) )
21	19, 20	syl5ibrcom 222	. . . . . . 7 |- ( U e. V -> ( ( E ` ( `' E ` { U , M } ) ) = { U , M } -> U e. ( E ` ( `' E ` { U , M } ) ) ) )
22	21	3ad2ant2 1010	. . . . . 6 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( ( E ` ( `' E ` { U , M } ) ) = { U , M } -> U e. ( E ` ( `' E ` { U , M } ) ) ) )
23	5, 22	mpd 15	. . . . 5 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> U e. ( E ` ( `' E ` { U , M } ) ) )
24		fveq2 5800	. . . . . . 7 |- ( i = ( `' E ` { U , M } ) -> ( E ` i ) = ( E ` ( `' E ` { U , M } ) ) )
25	24	eleq2d 2524	. . . . . 6 |- ( i = ( `' E ` { U , M } ) -> ( U e. ( E ` i ) <-> U e. ( E ` ( `' E ` { U , M } ) ) ) )
26	25	elrab 3224	. . . . 5 |- ( ( `' E ` { U , M } ) e. { i e. dom E | U e. ( E ` i ) } <-> ( ( `' E ` { U , M } ) e. dom E /\ U e. ( E ` ( `' E ` { U , M } ) ) ) )
27	18, 23, 26	sylanbrc 664	. . . 4 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( `' E ` { U , M } ) e. { i e. dom E | U e. ( E ` i ) } )
28		nbgraf1o.i	. . . 4 |- I = { i e. dom E | U e. ( E ` i ) }
29	27, 28	syl6eleqr 2553	. . 3 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( `' E ` { U , M } ) e. I )
30		nbgraf1o.f	. . . . 5 |- F = ( n e. N |-> ( iota_ i e. I ( E ` i ) = { U , n } ) )
31	1, 28, 30	nbgraf1olem1 23503	. . . 4 |- ( ( ( V USGrph E /\ U e. V ) /\ M e. N ) -> E! i e. I ( E ` i ) = { U , M } )
32	31	3impa 1183	. . 3 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> E! i e. I ( E ` i ) = { U , M } )
33	24	eqeq1d 2456	. . . 4 |- ( i = ( `' E ` { U , M } ) -> ( ( E ` i ) = { U , M } <-> ( E ` ( `' E ` { U , M } ) ) = { U , M } ) )
34	33	riota2 6185	. . 3 |- ( ( ( `' E ` { U , M } ) e. I /\ E! i e. I ( E ` i ) = { U , M } ) -> ( ( E ` ( `' E ` { U , M } ) ) = { U , M } <-> ( iota_ i e. I ( E ` i ) = { U , M } ) = ( `' E ` { U , M } ) ) )
35	29, 32, 34	syl2anc 661	. 2 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( ( E ` ( `' E ` { U , M } ) ) = { U , M } <-> ( iota_ i e. I ( E ` i ) = { U , M } ) = ( `' E ` { U , M } ) ) )
36	5, 35	mpbid 210	1 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( iota_ i e. I ( E ` i ) = { U , M } ) = ( `' E ` { U , M } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 965   = wceq 1370   e. wcel 1758   E!wreu 2801   {crab 2803   {cpr 3988   <.cop 3992   class class class wbr 4401   |-> cmpt 4459   `'ccnv 4948   dom cdm 4949   ran crn 4950   -1-1-onto->wf1o 5526   ` cfv 5527   iota_crio 6161  (class class class)co 6201   USGrph cusg 23417   Neighbors cnbgra 23482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-hash 12222  df-usgra 23419  df-nbgra 23485

This theorem is referenced by:  nbgraf1olem4  23506  nbgraf1olem5  23507