Theorem injsubmefmnd 18789

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 3442	. . . . 5 ⊢ 𝑥 ∈ V
2		f1eq1 6719	. . . . 5 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1→𝐴 ↔ 𝑥:𝐴–1-1→𝐴))
3	1, 2	elab 3637	. . . 4 ⊢ (𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ 𝑥:𝐴–1-1→𝐴)
4		f1f 6724	. . . . 5 ⊢ (𝑥:𝐴–1-1→𝐴 → 𝑥:𝐴⟶𝐴)
5		sursubmefmnd.m	. . . . . 6 ⊢ 𝑀 = (EndoFMnd‘𝐴)
6		eqid 2729	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
7	5, 6	elefmndbas 18765	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑀) ↔ 𝑥:𝐴⟶𝐴))
8	4, 7	imbitrrid 246	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐴–1-1→𝐴 → 𝑥 ∈ (Base‘𝑀)))
9	3, 8	biimtrid 242	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} → 𝑥 ∈ (Base‘𝑀)))
10	9	ssrdv 3943	. 2 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ⊆ (Base‘𝑀))
11	5	efmndid 18780	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝑀))
12		resiexg 7852	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
13		f1oi 6806	. . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
14		f1of1 6767	. . . . 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 6719	. . . 4 ⊢ (ℎ = ( I ↾ 𝐴) → (ℎ:𝐴–1-1→𝐴 ↔ ( I ↾ 𝐴):𝐴–1-1→𝐴))
17	12, 15, 16	elabd 3639	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
18	11, 17	eqeltrrd 2829	. 2 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝑀) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
19		vex 3442	. . . . . 6 ⊢ 𝑦 ∈ V
20		f1eq1 6719	. . . . . 6 ⊢ (ℎ = 𝑦 → (ℎ:𝐴–1-1→𝐴 ↔ 𝑦:𝐴–1-1→𝐴))
21	19, 20	elab 3637	. . . . 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 6735	. . . . . . 7 ⊢ ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
24	23	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
25		f1f 6724	. . . . . . . . . . . 12 ⊢ (𝑦:𝐴–1-1→𝐴 → 𝑦:𝐴⟶𝐴)
26	4, 25	anim12i 613	. . . . . . . . . . 11 ⊢ ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥:𝐴⟶𝐴 ∧ 𝑦:𝐴⟶𝐴))
27	5, 6	elefmndbas 18765	. . . . . . . . . . . 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	imbitrrid 246	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))))
30	29	imp 406	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)))
31		eqid 2729	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
32	5, 6, 31	efmndov 18773	. . . . . . . . 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 2813	. . . . . . 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 7870	. . . . . . . 8 ⊢ (𝑥 ∘ 𝑦) ∈ V
36		f1eq1 6719	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∘ 𝑦) → (ℎ:𝐴–1-1→𝐴 ↔ (𝑥 ∘ 𝑦):𝐴–1-1→𝐴))
37	35, 36	elab 3637	. . . . . . 7 ⊢ ((𝑥 ∘ 𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
38	34, 37	bitrdi 287	. . . . . 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 257	. . . . 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 412	. . . 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 242	. . 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 3170	. 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 18781	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
44		eqid 2729	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
45	6, 44, 31	issubm 18695	. . 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 1343	1 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  Vcvv 3438   ⊆ wss 3905   I cid 5517   ↾ cres 5625   ∘ ccom 5627  ⟶wf 6482  –1-1→wf1 6483  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  +_gcplusg 17179  0_gc0g 17361  Mndcmnd 18626  SubMndcsubmnd 18674  EndoFMndcefmnd 18760

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17139  df-plusg 17192  df-tset 17198  df-0g 17363  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-submnd 18676  df-efmnd 18761

This theorem is referenced by:  symgsubmefmnd  19295