Theorem frgrwopreglem5lem 27184

Description: Lemma for frgrwopreglem5 27185. (Contributed by AV, 5-Feb-2022.)

Hypotheses

Ref	Expression
frgrwopreg.v	|- V = (Vtx ` G )
frgrwopreg.d	|- D = (VtxDeg ` G )
frgrwopreg.a	|- A = { x e. V | ( D ` x ) = K }
frgrwopreg.b	|- B = ( V \ A )
frgrwopreg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
frgrwopreglem5lem	|- ( ( ( a e. A /\ x e. A ) /\ ( b e. B /\ y e. B ) ) -> ( ( D ` a ) = ( D ` x ) /\ ( D ` a ) =/= ( D ` b ) /\ ( D ` x ) =/= ( D ` y ) ) )

Distinct variable groups:   x, V   x, A   x, G   x, K   x, D   A, b   x, B   y, D   G, a, b, y, x   y, V

Allowed substitution hints:   A( y, a)   B( y, a, b)   D( a, b)   E( x, y, a, b)   K( y, a, b)   V( a, b)

Proof of Theorem frgrwopreglem5lem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrwopreg.a	. . . . . 6 |- A = { x e. V | ( D ` x ) = K }
2	1	rabeq2i 3197	. . . . 5 |- ( x e. A <-> ( x e. V /\ ( D ` x ) = K ) )
3		fveq2 6191	. . . . . . . 8 |- ( x = a -> ( D ` x ) = ( D ` a ) )
4	3	eqeq1d 2624	. . . . . . 7 |- ( x = a -> ( ( D ` x ) = K <-> ( D ` a ) = K ) )
5	4, 1	elrab2 3366	. . . . . 6 |- ( a e. A <-> ( a e. V /\ ( D ` a ) = K ) )
6		eqtr3 2643	. . . . . . . . 9 |- ( ( ( D ` a ) = K /\ ( D ` x ) = K ) -> ( D ` a ) = ( D ` x ) )
7	6	expcom 451	. . . . . . . 8 |- ( ( D ` x ) = K -> ( ( D ` a ) = K -> ( D ` a ) = ( D ` x ) ) )
8	7	adantl 482	. . . . . . 7 |- ( ( x e. V /\ ( D ` x ) = K ) -> ( ( D ` a ) = K -> ( D ` a ) = ( D ` x ) ) )
9	8	com12 32	. . . . . 6 |- ( ( D ` a ) = K -> ( ( x e. V /\ ( D ` x ) = K ) -> ( D ` a ) = ( D ` x ) ) )
10	5, 9	simplbiim 659	. . . . 5 |- ( a e. A -> ( ( x e. V /\ ( D ` x ) = K ) -> ( D ` a ) = ( D ` x ) ) )
11	2, 10	syl5bi 232	. . . 4 |- ( a e. A -> ( x e. A -> ( D ` a ) = ( D ` x ) ) )
12	11	imp 445	. . 3 |- ( ( a e. A /\ x e. A ) -> ( D ` a ) = ( D ` x ) )
13	12	adantr 481	. 2 |- ( ( ( a e. A /\ x e. A ) /\ ( b e. B /\ y e. B ) ) -> ( D ` a ) = ( D ` x ) )
14		frgrwopreg.v	. . . 4 |- V = (Vtx ` G )
15		frgrwopreg.d	. . . 4 |- D = (VtxDeg ` G )
16		frgrwopreg.b	. . . 4 |- B = ( V \ A )
17	14, 15, 1, 16	frgrwopreglem3 27178	. . 3 |- ( ( a e. A /\ b e. B ) -> ( D ` a ) =/= ( D ` b ) )
18	17	ad2ant2r 783	. 2 |- ( ( ( a e. A /\ x e. A ) /\ ( b e. B /\ y e. B ) ) -> ( D ` a ) =/= ( D ` b ) )
19		fveq2 6191	. . . . . . 7 |- ( x = z -> ( D ` x ) = ( D ` z ) )
20	19	eqeq1d 2624	. . . . . 6 |- ( x = z -> ( ( D ` x ) = K <-> ( D ` z ) = K ) )
21	20	cbvrabv 3199	. . . . 5 |- { x e. V | ( D ` x ) = K } = { z e. V | ( D ` z ) = K }
22	1, 21	eqtri 2644	. . . 4 |- A = { z e. V | ( D ` z ) = K }
23	14, 15, 22, 16	frgrwopreglem3 27178	. . 3 |- ( ( x e. A /\ y e. B ) -> ( D ` x ) =/= ( D ` y ) )
24	23	ad2ant2l 782	. 2 |- ( ( ( a e. A /\ x e. A ) /\ ( b e. B /\ y e. B ) ) -> ( D ` x ) =/= ( D ` y ) )
25	13, 18, 24	3jca 1242	1 |- ( ( ( a e. A /\ x e. A ) /\ ( b e. B /\ y e. B ) ) -> ( ( D ` a ) = ( D ` x ) /\ ( D ` a ) =/= ( D ` b ) /\ ( D ` x ) =/= ( D ` y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   \ cdif 3571   ` cfv 5888  Vtxcvtx 25874  Edgcedg 25939  VtxDegcvtxdg 26361

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-ral 2917  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:  frgrwopreglem5  27185