Theorem upgr1wlkdlem1 27005

Description: Lemma 1 for upgr1wlkd 27007. (Contributed by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
upgr1wlkd.p	|- P = <" X Y ">
upgr1wlkd.f	|- F = <" J ">
upgr1wlkd.x	|- ( ph -> X e. (Vtx ` G ) )
upgr1wlkd.y	|- ( ph -> Y e. (Vtx ` G ) )
upgr1wlkd.j	|- ( ph -> ( (iEdg ` G ) ` J ) = { X , Y } )

Assertion

Ref	Expression
upgr1wlkdlem1	|- ( ( ph /\ X = Y ) -> ( (iEdg ` G ) ` J ) = { X } )

Proof of Theorem upgr1wlkdlem1

Step	Hyp	Ref	Expression
1		upgr1wlkd.j	. . 3 |- ( ph -> ( (iEdg ` G ) ` J ) = { X , Y } )
2		preq2 4269	. . . . . . 7 |- ( Y = X -> { X , Y } = { X , X } )
3	2	eqeq2d 2632	. . . . . 6 |- ( Y = X -> ( ( (iEdg ` G ) ` J ) = { X , Y } <-> ( (iEdg ` G ) ` J ) = { X , X } ) )
4	3	eqcoms 2630	. . . . 5 |- ( X = Y -> ( ( (iEdg ` G ) ` J ) = { X , Y } <-> ( (iEdg ` G ) ` J ) = { X , X } ) )
5		simpl 473	. . . . . . 7 |- ( ( ( (iEdg ` G ) ` J ) = { X , X } /\ ph ) -> ( (iEdg ` G ) ` J ) = { X , X } )
6		dfsn2 4190	. . . . . . 7 |- { X } = { X , X }
7	5, 6	syl6eqr 2674	. . . . . 6 |- ( ( ( (iEdg ` G ) ` J ) = { X , X } /\ ph ) -> ( (iEdg ` G ) ` J ) = { X } )
8	7	ex 450	. . . . 5 |- ( ( (iEdg ` G ) ` J ) = { X , X } -> ( ph -> ( (iEdg ` G ) ` J ) = { X } ) )
9	4, 8	syl6bi 243	. . . 4 |- ( X = Y -> ( ( (iEdg ` G ) ` J ) = { X , Y } -> ( ph -> ( (iEdg ` G ) ` J ) = { X } ) ) )
10	9	com13 88	. . 3 |- ( ph -> ( ( (iEdg ` G ) ` J ) = { X , Y } -> ( X = Y -> ( (iEdg ` G ) ` J ) = { X } ) ) )
11	1, 10	mpd 15	. 2 |- ( ph -> ( X = Y -> ( (iEdg ` G ) ` J ) = { X } ) )
12	11	imp 445	1 |- ( ( ph /\ X = Y ) -> ( (iEdg ` G ) ` J ) = { X } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   {cpr 4179   ` cfv 5888   <"cs1 13294   <"cs2 13586  Vtxcvtx 25874  iEdgciedg 25875

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  upgr1wlkd  27007  upgr1trld  27008  upgr1pthd  27009  upgr1pthond  27010