Theorem 0wlkonlem2 26980

Description: Lemma 2 for 0wlkon 26981 and 0trlon 26985. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)

Hypothesis

Ref	Expression
0wlk.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
0wlkonlem2	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> P e. ( V ^pm ( 0 ... 0 ) ) )

Proof of Theorem 0wlkonlem2

Step	Hyp	Ref	Expression
1		ovex 6678	. 2 |- ( 0 ... 0 ) e. _V
2		0wlk.v	. . 3 |- V = (Vtx ` G )
3		fvex 6201	. . 3 |- (Vtx ` G ) e. _V
4	2, 3	eqeltri 2697	. 2 |- V e. _V
5		simpl 473	. 2 |- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> P : ( 0 ... 0 ) --> V )
6		fpmg 7883	. 2 |- ( ( ( 0 ... 0 ) e. _V /\ V e. _V /\ P : ( 0 ... 0 ) --> V ) -> P e. ( V ^pm ( 0 ... 0 ) ) )
7	1, 4, 5, 6	mp3an12i 1428	1 |- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> P e. ( V ^pm ( 0 ... 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   0cc0 9936   ...cfz 12326  Vtxcvtx 25874

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860

This theorem is referenced by:  0wlkon  26981  0trlon  26985  0pthon  26988