Theorem fusgreg2wsplem 27197

Description: Lemma for fusgreg2wsp 27200 and related theorems. (Contributed by AV, 8-Jan-2022.)

Hypotheses

Ref	Expression
frgrhash2wsp.v	|- V = (Vtx ` G )
fusgreg2wsp.m	|- M = ( a e. V |-> { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = a } )

Assertion

Ref	Expression
fusgreg2wsplem	|- ( N e. V -> ( p e. ( M ` N ) <-> ( p e. ( 2 WSPathsN G ) /\ ( p ` 1 ) = N ) ) )

Distinct variable groups:   G, a   V, a   w, G   N, a, w   w, p

Allowed substitution hints:   G( p)   M( w, p, a)   N( p)   V( w, p)

Proof of Theorem fusgreg2wsplem

Step	Hyp	Ref	Expression
1		eqeq2 2633	. . . . 5 |- ( a = N -> ( ( w ` 1 ) = a <-> ( w ` 1 ) = N ) )
2	1	rabbidv 3189	. . . 4 |- ( a = N -> { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = a } = { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = N } )
3		fusgreg2wsp.m	. . . 4 |- M = ( a e. V |-> { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = a } )
4		ovex 6678	. . . . 5 |- ( 2 WSPathsN G ) e. _V
5	4	rabex 4813	. . . 4 |- { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = N } e. _V
6	2, 3, 5	fvmpt 6282	. . 3 |- ( N e. V -> ( M ` N ) = { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = N } )
7	6	eleq2d 2687	. 2 |- ( N e. V -> ( p e. ( M ` N ) <-> p e. { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = N } ) )
8		fveq1 6190	. . . 4 |- ( w = p -> ( w ` 1 ) = ( p ` 1 ) )
9	8	eqeq1d 2624	. . 3 |- ( w = p -> ( ( w ` 1 ) = N <-> ( p ` 1 ) = N ) )
10	9	elrab 3363	. 2 |- ( p e. { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = N } <-> ( p e. ( 2 WSPathsN G ) /\ ( p ` 1 ) = N ) )
11	7, 10	syl6bb 276	1 |- ( N e. V -> ( p e. ( M ` N ) <-> ( p e. ( 2 WSPathsN G ) /\ ( p ` 1 ) = N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   2c2 11070  Vtxcvtx 25874   WSPathsN cwwspthsn 26720

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  fusgr2wsp2nb  27198  fusgreg2wsp  27200  2wspmdisj  27201