Theorem uhgr3cyclexlem 27041

Description: Lemma for uhgr3cyclex 27042. (Contributed by AV, 12-Feb-2021.)

Hypotheses

Ref	Expression
uhgr3cyclex.v	|- V = (Vtx ` G )
uhgr3cyclex.e	|- E = (Edg ` G )
uhgr3cyclex.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
uhgr3cyclexlem	|- ( ( ( ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( J e. dom I /\ { B , C } = ( I ` J ) ) /\ ( K e. dom I /\ { C , A } = ( I ` K ) ) ) ) -> J =/= K )

Proof of Theorem uhgr3cyclexlem

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . . 9 |- ( J = K -> ( I ` J ) = ( I ` K ) )
2	1	eqeq2d 2632	. . . . . . . 8 |- ( J = K -> ( { B , C } = ( I ` J ) <-> { B , C } = ( I ` K ) ) )
3		eqeq2 2633	. . . . . . . . . . . 12 |- ( ( I ` K ) = { C , A } -> ( { B , C } = ( I ` K ) <-> { B , C } = { C , A } ) )
4	3	eqcoms 2630	. . . . . . . . . . 11 |- ( { C , A } = ( I ` K ) -> ( { B , C } = ( I ` K ) <-> { B , C } = { C , A } ) )
5		prcom 4267	. . . . . . . . . . . . . 14 |- { C , A } = { A , C }
6	5	eqeq1i 2627	. . . . . . . . . . . . 13 |- ( { C , A } = { B , C } <-> { A , C } = { B , C } )
7		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( A e. V /\ B e. V ) -> A e. V )
8		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( A e. V /\ B e. V ) -> B e. V )
9	7, 8	preq1b 4377	. . . . . . . . . . . . . 14 |- ( ( A e. V /\ B e. V ) -> ( { A , C } = { B , C } <-> A = B ) )
10	9	biimpcd 239	. . . . . . . . . . . . 13 |- ( { A , C } = { B , C } -> ( ( A e. V /\ B e. V ) -> A = B ) )
11	6, 10	sylbi 207	. . . . . . . . . . . 12 |- ( { C , A } = { B , C } -> ( ( A e. V /\ B e. V ) -> A = B ) )
12	11	eqcoms 2630	. . . . . . . . . . 11 |- ( { B , C } = { C , A } -> ( ( A e. V /\ B e. V ) -> A = B ) )
13	4, 12	syl6bi 243	. . . . . . . . . 10 |- ( { C , A } = ( I ` K ) -> ( { B , C } = ( I ` K ) -> ( ( A e. V /\ B e. V ) -> A = B ) ) )
14	13	adantl 482	. . . . . . . . 9 |- ( ( K e. dom I /\ { C , A } = ( I ` K ) ) -> ( { B , C } = ( I ` K ) -> ( ( A e. V /\ B e. V ) -> A = B ) ) )
15	14	com12 32	. . . . . . . 8 |- ( { B , C } = ( I ` K ) -> ( ( K e. dom I /\ { C , A } = ( I ` K ) ) -> ( ( A e. V /\ B e. V ) -> A = B ) ) )
16	2, 15	syl6bi 243	. . . . . . 7 |- ( J = K -> ( { B , C } = ( I ` J ) -> ( ( K e. dom I /\ { C , A } = ( I ` K ) ) -> ( ( A e. V /\ B e. V ) -> A = B ) ) ) )
17	16	adantld 483	. . . . . 6 |- ( J = K -> ( ( J e. dom I /\ { B , C } = ( I ` J ) ) -> ( ( K e. dom I /\ { C , A } = ( I ` K ) ) -> ( ( A e. V /\ B e. V ) -> A = B ) ) ) )
18	17	com14 96	. . . . 5 |- ( ( A e. V /\ B e. V ) -> ( ( J e. dom I /\ { B , C } = ( I ` J ) ) -> ( ( K e. dom I /\ { C , A } = ( I ` K ) ) -> ( J = K -> A = B ) ) ) )
19	18	imp32 449	. . . 4 |- ( ( ( A e. V /\ B e. V ) /\ ( ( J e. dom I /\ { B , C } = ( I ` J ) ) /\ ( K e. dom I /\ { C , A } = ( I ` K ) ) ) ) -> ( J = K -> A = B ) )
20	19	necon3d 2815	. . 3 |- ( ( ( A e. V /\ B e. V ) /\ ( ( J e. dom I /\ { B , C } = ( I ` J ) ) /\ ( K e. dom I /\ { C , A } = ( I ` K ) ) ) ) -> ( A =/= B -> J =/= K ) )
21	20	impancom 456	. 2 |- ( ( ( A e. V /\ B e. V ) /\ A =/= B ) -> ( ( ( J e. dom I /\ { B , C } = ( I ` J ) ) /\ ( K e. dom I /\ { C , A } = ( I ` K ) ) ) -> J =/= K ) )
22	21	imp 445	1 |- ( ( ( ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( J e. dom I /\ { B , C } = ( I ` J ) ) /\ ( K e. dom I /\ { C , A } = ( I ` K ) ) ) ) -> J =/= K )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {cpr 4179   dom cdm 5114   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  uhgr3cyclex  27042