Theorem injsubmefmnd 18946

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 3461	. . . . 5 ⊢ 𝑥 ∈ V
2		f1eq1 6759	. . . . 5 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1→𝐴 ↔ 𝑥:𝐴–1-1→𝐴))
3	1, 2	elab 3641	. . . 4 ⊢ (𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ 𝑥:𝐴–1-1→𝐴)
4		f1f 6764	. . . . 5 ⊢ (𝑥:𝐴–1-1→𝐴 → 𝑥:𝐴⟶𝐴)
5		sursubmefmnd.m	. . . . . 6 ⊢ 𝑀 = (EndoFMnd‘𝐴)
6		eqid 2765	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
7	5, 6	elefmndbas 18922	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑀) ↔ 𝑥:𝐴⟶𝐴))
8	4, 7	imbitrrid 249	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐴–1-1→𝐴 → 𝑥 ∈ (Base‘𝑀)))
9	3, 8	biimtrid 245	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} → 𝑥 ∈ (Base‘𝑀)))
10	9	ssrdv 3945	. 2 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ⊆ (Base‘𝑀))
11	5	efmndid 18937	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝑀))
12		resiexg 7897	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
13		f1oi 6849	. . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
14		f1of1 6809	. . . . 5 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 → ( I ↾ 𝐴):𝐴–1-1→𝐴)
15	13, 14	mp1i 14	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴):𝐴–1-1→𝐴)
16		f1eq1 6759	. . . 4 ⊢ (ℎ = ( I ↾ 𝐴) → (ℎ:𝐴–1-1→𝐴 ↔ ( I ↾ 𝐴):𝐴–1-1→𝐴))
17	12, 15, 16	elabd 3643	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
18	11, 17	eqeltrrd 2866	. 2 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝑀) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
19		vex 3461	. . . . . 6 ⊢ 𝑦 ∈ V
20		f1eq1 6759	. . . . . 6 ⊢ (ℎ = 𝑦 → (ℎ:𝐴–1-1→𝐴 ↔ 𝑦:𝐴–1-1→𝐴))
21	19, 20	elab 3641	. . . . 5 ⊢ (𝑦 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ 𝑦:𝐴–1-1→𝐴)
22	3, 21	anbi12i 639	. . . 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 6777	. . . . . . 7 ⊢ ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
24	23	adantl 486	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
25		f1f 6764	. . . . . . . . . . . 12 ⊢ (𝑦:𝐴–1-1→𝐴 → 𝑦:𝐴⟶𝐴)
26	4, 25	anim12i 624	. . . . . . . . . . 11 ⊢ ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥:𝐴⟶𝐴 ∧ 𝑦:𝐴⟶𝐴))
27	5, 6	elefmndbas 18922	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ (Base‘𝑀) ↔ 𝑦:𝐴⟶𝐴))
28	7, 27	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) ↔ (𝑥:𝐴⟶𝐴 ∧ 𝑦:𝐴⟶𝐴)))
29	26, 28	imbitrrid 249	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))))
30	29	imp 411	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)))
31		eqid 2765	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
32	5, 6, 31	efmndov 18930	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ∘ 𝑦))
33	30, 32	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ∘ 𝑦))
34	33	eleq1d 2850	. . . . . . 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 7915	. . . . . . . 8 ⊢ (𝑥 ∘ 𝑦) ∈ V
36		f1eq1 6759	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∘ 𝑦) → (ℎ:𝐴–1-1→𝐴 ↔ (𝑥 ∘ 𝑦):𝐴–1-1→𝐴))
37	35, 36	elab 3641	. . . . . . 7 ⊢ ((𝑥 ∘ 𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
38	34, 37	bitrdi 290	. . . . . 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 260	. . . . 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 417	. . . 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	biimtrid 245	. . 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 3206	. 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 18938	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
44		eqid 2765	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
45	6, 44, 31	issubm 18851	. . 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 18	. 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 1359	1 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {cab 2743  ∀wral 3079  Vcvv 3457   ⊆ wss 3907   I cid 5546   ↾ cres 5654   ∘ ccom 5656  ⟶wf 6521  –1-1→wf1 6522  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  +_gcplusg 17300  0_gc0g 17482  Mndcmnd 18782  SubMndcsubmnd 18830  EndoFMndcefmnd 18917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-z 12583  df-uz 12854  df-fz 13527  df-struct 17197  df-slot 17232  df-ndx 17244  df-base 17260  df-plusg 17313  df-tset 17319  df-0g 17484  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-submnd 18832  df-efmnd 18918

This theorem is referenced by:  symgsubmefmnd  19459