Theorem nbgraf1olem3 25164

Description: Lemma 3 for nbgraf1o 25168. 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 2520	. . . 4 |- ( M e. N <-> M e. ( <. V , E >. Neighbors U ) )
3		nbgracnvfv 25161	. . . 4 |- ( ( V USGrph E /\ M e. ( <. V , E >. Neighbors U ) ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
4	2, 3	sylan2b 478	. . 3 |- ( ( V USGrph E /\ M e. N ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
5	4	3adant2 1026	. 2 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
6		usgraf1o 25078	. . . . . . 7 |- ( V USGrph E -> E : dom E -1-1-onto-> ran E )
7	6	3ad2ant1 1028	. . . . . 6 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> E : dom E -1-1-onto-> ran E )
8		nbgraeledg 25151	. . . . . . . . . . . 12 |- ( V USGrph E -> ( M e. ( <. V , E >. Neighbors U ) <-> { M , U } e. ran E ) )
9		prcom 4049	. . . . . . . . . . . . 13 |- { M , U } = { U , M }
10	9	eleq1i 2519	. . . . . . . . . . . 12 |- ( { M , U } e. ran E <-> { U , M } e. ran E )
11	8, 10	syl6bb 265	. . . . . . . . . . 11 |- ( V USGrph E -> ( M e. ( <. V , E >. Neighbors U ) <-> { U , M } e. ran E ) )
12	11	biimpcd 228	. . . . . . . . . 10 |- ( M e. ( <. V , E >. Neighbors U ) -> ( V USGrph E -> { U , M } e. ran E ) )
13	12	a1d 26	. . . . . . . . 9 |- ( M e. ( <. V , E >. Neighbors U ) -> ( U e. V -> ( V USGrph E -> { U , M } e. ran E ) ) )
14	13, 1	eleq2s 2546	. . . . . . . 8 |- ( M e. N -> ( U e. V -> ( V USGrph E -> { U , M } e. ran E ) ) )
15	14	com13 83	. . . . . . 7 |- ( V USGrph E -> ( U e. V -> ( M e. N -> { U , M } e. ran E ) ) )
16	15	3imp 1201	. . . . . 6 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> { U , M } e. ran E )
17		f1ocnvdm 6181	. . . . . 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 666	. . . . 5 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( `' E ` { U , M } ) e. dom E )
19		prid1g 4077	. . . . . . . 8 |- ( U e. V -> U e. { U , M } )
20		eleq2 2517	. . . . . . . 8 |- ( ( E ` ( `' E ` { U , M } ) ) = { U , M } -> ( U e. ( E ` ( `' E ` { U , M } ) ) <-> U e. { U , M } ) )
21	19, 20	syl5ibrcom 226	. . . . . . 7 |- ( U e. V -> ( ( E ` ( `' E ` { U , M } ) ) = { U , M } -> U e. ( E ` ( `' E ` { U , M } ) ) ) )
22	21	3ad2ant2 1029	. . . . . 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 5863	. . . . . . 7 |- ( i = ( `' E ` { U , M } ) -> ( E ` i ) = ( E ` ( `' E ` { U , M } ) ) )
25	24	eleq2d 2513	. . . . . 6 |- ( i = ( `' E ` { U , M } ) -> ( U e. ( E ` i ) <-> U e. ( E ` ( `' E ` { U , M } ) ) ) )
26	25	elrab 3195	. . . . 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 669	. . . 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 2539	. . 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 25162	. . . 4 |- ( ( ( V USGrph E /\ U e. V ) /\ M e. N ) -> E! i e. I ( E ` i ) = { U , M } )
32	31	3impa 1202	. . 3 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> E! i e. I ( E ` i ) = { U , M } )
33	24	eqeq1d 2452	. . . 4 |- ( i = ( `' E ` { U , M } ) -> ( ( E ` i ) = { U , M } <-> ( E ` ( `' E ` { U , M } ) ) = { U , M } ) )
34	33	riota2 6272	. . 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 666	. 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 214	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 188   /\ w3a 984   = wceq 1443   e. wcel 1886   E!wreu 2738   {crab 2740   {cpr 3969   <.cop 3973   class class class wbr 4401   |-> cmpt 4460   `'ccnv 4832   dom cdm 4833   ran crn 4834   -1-1-onto->wf1o 5580   ` cfv 5581   iota_crio 6249  (class class class)co 6288   USGrph cusg 25050   Neighbors cnbgra 25138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-hash 12513  df-usgra 25053  df-nbgra 25141

This theorem is referenced by:  nbgraf1olem4  25165  nbgraf1olem5  25166