Theorem injsubmefmnd 18914

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 3457	. . . . 5 ⊢ 𝑥 ∈ V
2		f1eq1 6751	. . . . 5 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1→𝐴 ↔ 𝑥:𝐴–1-1→𝐴))
3	1, 2	elab 3638	. . . 4 ⊢ (𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ 𝑥:𝐴–1-1→𝐴)
4		f1f 6756	. . . . 5 ⊢ (𝑥:𝐴–1-1→𝐴 → 𝑥:𝐴⟶𝐴)
5		sursubmefmnd.m	. . . . . 6 ⊢ 𝑀 = (EndoFMnd‘𝐴)
6		eqid 2761	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
7	5, 6	elefmndbas 18890	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑀) ↔ 𝑥:𝐴⟶𝐴))
8	4, 7	imbitrrid 248	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐴–1-1→𝐴 → 𝑥 ∈ (Base‘𝑀)))
9	3, 8	biimtrid 244	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} → 𝑥 ∈ (Base‘𝑀)))
10	9	ssrdv 3942	. 2 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ⊆ (Base‘𝑀))
11	5	efmndid 18905	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝑀))
12		resiexg 7889	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
13		f1oi 6841	. . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
14		f1of1 6801	. . . . 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 6751	. . . 4 ⊢ (ℎ = ( I ↾ 𝐴) → (ℎ:𝐴–1-1→𝐴 ↔ ( I ↾ 𝐴):𝐴–1-1→𝐴))
17	12, 15, 16	elabd 3640	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
18	11, 17	eqeltrrd 2862	. 2 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝑀) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴})
19		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
20		f1eq1 6751	. . . . . 6 ⊢ (ℎ = 𝑦 → (ℎ:𝐴–1-1→𝐴 ↔ 𝑦:𝐴–1-1→𝐴))
21	19, 20	elab 3638	. . . . 5 ⊢ (𝑦 ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ 𝑦:𝐴–1-1→𝐴)
22	3, 21	anbi12i 637	. . . 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 6769	. . . . . . 7 ⊢ ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
24	23	adantl 485	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
25		f1f 6756	. . . . . . . . . . . 12 ⊢ (𝑦:𝐴–1-1→𝐴 → 𝑦:𝐴⟶𝐴)
26	4, 25	anim12i 622	. . . . . . . . . . 11 ⊢ ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥:𝐴⟶𝐴 ∧ 𝑦:𝐴⟶𝐴))
27	5, 6	elefmndbas 18890	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ (Base‘𝑀) ↔ 𝑦:𝐴⟶𝐴))
28	7, 27	anbi12d 641	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) ↔ (𝑥:𝐴⟶𝐴 ∧ 𝑦:𝐴⟶𝐴)))
29	26, 28	imbitrrid 248	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))))
30	29	imp 410	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴–1-1→𝐴 ∧ 𝑦:𝐴–1-1→𝐴)) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)))
31		eqid 2761	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
32	5, 6, 31	efmndov 18898	. . . . . . . . 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 2846	. . . . . . 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 7907	. . . . . . . 8 ⊢ (𝑥 ∘ 𝑦) ∈ V
36		f1eq1 6751	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∘ 𝑦) → (ℎ:𝐴–1-1→𝐴 ↔ (𝑥 ∘ 𝑦):𝐴–1-1→𝐴))
37	35, 36	elab 3638	. . . . . . 7 ⊢ ((𝑥 ∘ 𝑦) ∈ {ℎ ∣ ℎ:𝐴–1-1→𝐴} ↔ (𝑥 ∘ 𝑦):𝐴–1-1→𝐴)
38	34, 37	bitrdi 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 416	. . . 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 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 3202	. 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 18906	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
44		eqid 2761	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
45	6, 44, 31	issubm 18820	. . 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 1355	1 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  Vcvv 3453   ⊆ wss 3904   I cid 5539   ↾ cres 5647   ∘ ccom 5649  ⟶wf 6513  –1-1→wf1 6514  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  0_gc0g 17451  Mndcmnd 18751  SubMndcsubmnd 18799  EndoFMndcefmnd 18885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-plusg 17282  df-tset 17288  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-submnd 18801  df-efmnd 18886

This theorem is referenced by:  symgsubmefmnd  19421