Metamath Proof Explorer


Theorem symgpssefmnd

Description: For a set A with more than one element, the symmetric group on A is a proper subset of the monoid of endofunctions on A . (Contributed by AV, 31-Mar-2024)

Ref Expression
Hypotheses symgpssefmnd.m No typesetting found for |- M = ( EndoFMnd ` A ) with typecode |-
symgpssefmnd.g G = SymGrp A
Assertion symgpssefmnd A V 1 < A Base G Base M

Proof

Step Hyp Ref Expression
1 symgpssefmnd.m Could not format M = ( EndoFMnd ` A ) : No typesetting found for |- M = ( EndoFMnd ` A ) with typecode |-
2 symgpssefmnd.g G = SymGrp A
3 hashgt12el A V 1 < A x A y A x y
4 eqid Base G = Base G
5 2 4 symgbasmap x Base G x A A
6 eqid Base M = Base M
7 1 6 efmndbas Base M = A A
8 5 7 eleqtrrdi x Base G x Base M
9 8 ssriv Base G Base M
10 9 a1i A V x A y A x y Base G Base M
11 fconst6g x A A × x : A A
12 11 adantr x A y A A × x : A A
13 12 3ad2ant2 A V x A y A x y A × x : A A
14 1 6 elefmndbas A V A × x Base M A × x : A A
15 14 3ad2ant1 A V x A y A x y A × x Base M A × x : A A
16 13 15 mpbird A V x A y A x y A × x Base M
17 fconstg x A A × x : A x
18 17 adantr x A y A A × x : A x
19 18 3ad2ant2 A V x A y A x y A × x : A x
20 id x A y A x y x A y A x y
21 20 3expa x A y A x y x A y A x y
22 21 3adant1 A V x A y A x y x A y A x y
23 nf1oconst A × x : A x x A y A x y ¬ A × x : A 1-1 onto A
24 19 22 23 syl2anc A V x A y A x y ¬ A × x : A 1-1 onto A
25 2 4 elsymgbas A V A × x Base G A × x : A 1-1 onto A
26 25 notbid A V ¬ A × x Base G ¬ A × x : A 1-1 onto A
27 26 3ad2ant1 A V x A y A x y ¬ A × x Base G ¬ A × x : A 1-1 onto A
28 24 27 mpbird A V x A y A x y ¬ A × x Base G
29 10 16 28 ssnelpssd A V x A y A x y Base G Base M
30 29 3exp A V x A y A x y Base G Base M
31 30 rexlimdvv A V x A y A x y Base G Base M
32 31 adantr A V 1 < A x A y A x y Base G Base M
33 3 32 mpd A V 1 < A Base G Base M