Theorem nbgraf1olem3 24648

Description: Lemma 3 for nbgraf1o 24652. 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 24645	. . . 4 |- ( ( V USGrph E /\ M e. ( <. V , E >. Neighbors U ) ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
4	2, 3	sylan2b 473	. . 3 |- ( ( V USGrph E /\ M e. N ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
5	4	3adant2 1013	. 2 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( E ` ( `' E ` { U , M } ) ) = { U , M } )
6		usgraf1o 24563	. . . . . . 7 |- ( V USGrph E -> E : dom E -1-1-onto-> ran E )
7	6	3ad2ant1 1015	. . . . . 6 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> E : dom E -1-1-onto-> ran E )
8		nbgraeledg 24635	. . . . . . . . . . . 12 |- ( V USGrph E -> ( M e. ( <. V , E >. Neighbors U ) <-> { M , U } e. ran E ) )
9		prcom 4094	. . . . . . . . . . . . 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 1188	. . . . . 6 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> { U , M } e. ran E )
17		f1ocnvdm 6163	. . . . . 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 659	. . . . 5 |- ( ( V USGrph E /\ U e. V /\ M e. N ) -> ( `' E ` { U , M } ) e. dom E )
19		prid1g 4122	. . . . . . . 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 1016	. . . . . 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 5848	. . . . . . 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 3254	. . . . 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 662	. . . 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 24646	. . . 4 |- ( ( ( V USGrph E /\ U e. V ) /\ M e. N ) -> E! i e. I ( E ` i ) = { U , M } )
32	31	3impa 1189	. . 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 6254	. . 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 659	. 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 971   = wceq 1398   e. wcel 1823   E!wreu 2806   {crab 2808   {cpr 4018   <.cop 4022   class class class wbr 4439   |-> cmpt 4497   `'ccnv 4987   dom cdm 4988   ran crn 4989   -1-1-onto->wf1o 5569   ` cfv 5570   iota_crio 6231  (class class class)co 6270   USGrph cusg 24535   Neighbors cnbgra 24622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12391  df-usgra 24538  df-nbgra 24625

This theorem is referenced by:  nbgraf1olem4  24649  nbgraf1olem5  24650