Theorem nbgraf1olem3 24116

Description: Lemma 3 for nbgraf1o 24120. 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 2545	. . . 4 |- ( M e. N <-> M e. ( <. V , E >. Neighbors U ) )
3		nbgracnvfv 24113	. . . 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 1015	. 2 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
6		usgraf1o 24031	. . . . . . 7 |- ( V USGrph E -> E : dom E -1-1-onto-> ran E )
7	6	3ad2ant1 1017	. . . . . 6 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> E : dom E -1-1-onto-> ran E )
8		nbgraeledg 24103	. . . . . . . . . . . 12 |- ( V USGrph E -> ( M e. ( <. V , E >. Neighbors U ) <-> { M , U } e. ran E ) )
9		prcom 4105	. . . . . . . . . . . . 13 |- { M , U } = { U , M }
10	9	eleq1i 2544	. . . . . . . . . . . 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 2575	. . . . . . . 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 1190	. . . . . 6 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> { U , M } e. ran E )
17		f1ocnvdm 6174	. . . . . 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 4133	. . . . . . . 8 |- ( U e. V -> U e. { U , M } )
20		eleq2 2540	. . . . . . . 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 1018	. . . . . 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 5864	. . . . . . 7 |- ( i = ( `' E ` { U , M } ) -> ( E ` i ) = ( E ` ( `' E ` { U , M } ) ) )
25	24	eleq2d 2537	. . . . . 6 |- ( i = ( `' E ` { U , M } ) -> ( U e. ( E ` i ) <-> U e. ( E ` ( `' E ` { U , M } ) ) ) )
26	25	elrab 3261	. . . . 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 2566	. . 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 24114	. . . 4 |- ( ( ( V USGrph E /\ U e. V ) /\ M e. N ) -> E! i e. I ( E ` i ) = { U , M } )
32	31	3impa 1191	. . 3 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> E! i e. I ( E ` i ) = { U , M } )
33	24	eqeq1d 2469	. . . 4 |- ( i = ( `' E ` { U , M } ) -> ( ( E ` i ) = { U , M } <-> ( E ` ( `' E ` { U , M } ) ) = { U , M } ) )
34	33	riota2 6266	. . 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 973   = wceq 1379   e. wcel 1767   E!wreu 2816   {crab 2818   {cpr 4029   <.cop 4033   class class class wbr 4447   |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   -1-1-onto->wf1o 5585   ` cfv 5586   iota_crio 6242  (class class class)co 6282   USGrph cusg 24003   Neighbors cnbgra 24090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12368  df-usgra 24006  df-nbgra 24093

This theorem is referenced by:  nbgraf1olem4  24117  nbgraf1olem5  24118