Theorem injsubmefmnd 18057

Description: The set of injective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)

Hypothesis

Ref	Expression
sursubmefmnd.m	⊢ 𝑀 = (EndoFMnd‘𝐴)

Assertion

Ref	Expression
injsubmefmnd	⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀))

Distinct variable group:   𝐴,ℎ

Allowed substitution hints:   𝑀(ℎ)   𝑉(ℎ)

Proof of Theorem injsubmefmnd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3494	. . . . 5 ⊢ 𝑥 ∈ V
2		f1eq1 6563	. . . . 5 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1→𝐴 ↔ 𝑥:𝐴–1-1→𝐴))
3	1, 2	elab 3663	. . . 4 ⊢ (𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ 𝑥:𝐴–1-1→𝐴)
4		f1f 6568	. . . . 5 ⊢ (𝑥:𝐴–1-1→𝐴 → 𝑥:𝐴⟶𝐴)
5		sursubmefmnd.m	. . . . . 6 ⊢ 𝑀 = (EndoFMnd‘𝐴)
6		eqid 2820	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
7	5, 6	elefmndbas 18033	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑀) ↔ 𝑥:𝐴⟶𝐴))
8	4, 7	syl5ibr 248	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐴–1-1→𝐴 → 𝑥 ∈ (Base‘𝑀)))
9	3, 8	syl5bi 244	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} → 𝑥 ∈ (Base‘𝑀)))
10	9	ssrdv 3966	. 2 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ⊆ (Base‘𝑀))
11	5	efmndid 18048	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝑀))
12		resiexg 7612	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
13		f1oi 6645	. . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
14		f1of1 6607	. . . . 5 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 → ( I ↾ 𝐴):𝐴–1-1→𝐴)
15	13, 14	mp1i 13	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴):𝐴–1-1→𝐴)
16		f1eq1 6563	. . . 4 ⊢ (ℎ = ( I ↾ 𝐴) → (ℎ:𝐴–1-1→𝐴 ↔ ( I ↾ 𝐴):𝐴–1-1→𝐴))
17	12, 15, 16	elabd 3665	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
18	11, 17	eqeltrrd 2913	. 2 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝑀) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
19		vex 3494	. . . . . 6 ⊢ 𝑦 ∈ V
20		f1eq1 6563	. . . . . 6 ⊢ (ℎ = 𝑦 → (ℎ:𝐴–1-1→𝐴 ↔ 𝑦:𝐴–1-1→𝐴))
21	19, 20	elab 3663	. . . . 5 ⊢ (𝑦 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ 𝑦:𝐴–1-1→𝐴)
22	3, 21	anbi12i 628	. . . 4 ⊢ ((𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∧ 𝑦 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴}) ↔ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴))
23		f1co 6578	. . . . . . 7 ⊢ ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
24	23	adantl 484	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
25		f1f 6568	. . . . . . . . . . . 12 ⊢ (𝑦:𝐴–1-1→𝐴 → 𝑦:𝐴⟶𝐴)
26	4, 25	anim12i 614	. . . . . . . . . . 11 ⊢ ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥:𝐴⟶𝐴 ∧ 𝑦:𝐴⟶𝐴))
27	5, 6	elefmndbas 18033	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ (Base‘𝑀) ↔ 𝑦:𝐴⟶𝐴))
28	7, 27	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) ↔ (𝑥:𝐴⟶𝐴 ∧ 𝑦:𝐴⟶𝐴)))
29	26, 28	syl5ibr 248	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))))
30	29	imp 409	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)))
31		eqid 2820	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
32	5, 6, 31	efmndov 18041	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ∘ 𝑦))
33	30, 32	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ∘ 𝑦))
34	33	eleq1d 2896	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → ((𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ (𝑥 ∘ 𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴}))
35	1, 19	coex 7628	. . . . . . . 8 ⊢ (𝑥 ∘ 𝑦) ∈ V
36		f1eq1 6563	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∘ 𝑦) → (ℎ:𝐴–1-1→𝐴 ↔ (𝑥 ∘ 𝑦):𝐴–1-1→𝐴))
37	35, 36	elab 3663	. . . . . . 7 ⊢ ((𝑥 ∘ 𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
38	34, 37	syl6bb 289	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → ((𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ (𝑥 ∘ 𝑦):𝐴–1-1→𝐴))
39	24, 38	mpbird 259	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
40	39	ex 415	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴}))
41	22, 40	syl5bi 244	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∧ 𝑦 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴}) → (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴}))
42	41	ralrimivv 3189	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴}∀𝑦 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
43	5	efmndmnd 18049	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
44		eqid 2820	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
45	6, 44, 31	issubm 17963	. . 3 ⊢ (𝑀 ∈ Mnd → ({ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀) ↔ ({ℎ ∣ ℎ:𝐴–1-1→𝐴} ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∧ ∀𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴}∀𝑦 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})))
46	43, 45	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → ({ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀) ↔ ({ℎ ∣ ℎ:𝐴–1-1→𝐴} ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∧ ∀𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴}∀𝑦 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})))
47	10, 18, 42, 46	mpbir3and 1337	1 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113  {cab 2798  ∀wral 3137  Vcvv 3491   ⊆ wss 3929   I cid 5452   ↾ cres 5550   ∘ ccom 5552  ⟶wf 6344  –1-1→wf1 6345  –1-1-onto→wf1o 6347  ‘cfv 6348  (class class class)co 7149  Basecbs 16478  +_gcplusg 16560  0_gc0g 16708  Mndcmnd 17906  SubMndcsubmnd 17950  EndoFMndcefmnd 18028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-1o 8095  df-oadd 8099  df-er 8282  df-map 8401  df-en 8503  df-dom 8504  df-sdom 8505  df-fin 8506  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11632  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12890  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-plusg 16573  df-tset 16579  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-efmnd 18029

This theorem is referenced by:  symgsubmefmnd  18521