Theorem injsubmefmnd 19125

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⟶1-1𝐴} ∈ (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 3458	. . . . 5 ⊢ 𝑥 ∈ V
2		f1eq1 6798	. . . . 5 ⊢ (ℎ = 𝑥 → (ℎ:𝐴⟶1-1𝐴 ↔ 𝑥:𝐴⟶1-1𝐴))
3	1, 2	elab 3636	. . . 4 ⊢ (𝑥 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ↔ 𝑥:𝐴⟶1-1𝐴)
4		f1f 6803	. . . . 5 ⊢ (𝑥:𝐴⟶1-1𝐴 → 𝑥:𝐴⟶𝐴)
5		sursubmefmnd.m	. . . . . 6 ⊢ 𝑀 = (EndoFMnd‘𝐴)
6		eqid 2763	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
7	5, 6	elefmndbas 19101	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑀) ↔ 𝑥:𝐴⟶𝐴))
8	4, 7	imbitrrid 249	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐴⟶1-1𝐴 → 𝑥 ∈ (Base‘𝑀)))
9	3, 8	biimtrid 245	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} → 𝑥 ∈ (Base‘𝑀)))
10	9	ssrdv 3919	. 2 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ⊆ (Base‘𝑀))
11	5	efmndid 19116	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝑀))
12		resiexg 7951	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
13		f1oi 6889	. . . . 5 ⊢ ( I ↾ 𝐴):𝐴⟶1-1onto𝐴
14		f1of1 6848	. . . . 5 ⊢ (( I ↾ 𝐴):𝐴⟶1-1onto𝐴 → ( I ↾ 𝐴):𝐴⟶1-1𝐴)
15	13, 14	mp1i 14	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴):𝐴⟶1-1𝐴)
16		f1eq1 6798	. . . 4 ⊢ (ℎ = ( I ↾ 𝐴) → (ℎ:𝐴⟶1-1𝐴 ↔ ( I ↾ 𝐴):𝐴⟶1-1𝐴))
17	12, 15, 16	elabd 3638	. . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴})
18	11, 17	eqeltrrd 2865	. 2 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝑀) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴})
19		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
20		f1eq1 6798	. . . . . 6 ⊢ (ℎ = 𝑦 → (ℎ:𝐴⟶1-1𝐴 ↔ 𝑦:𝐴⟶1-1𝐴))
21	19, 20	elab 3636	. . . . 5 ⊢ (𝑦 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ↔ 𝑦:𝐴⟶1-1𝐴)
22	3, 21	anbi12i 640	. . . 4 ⊢ ((𝑥 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ∧ 𝑦 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴}) ↔ (𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴))
23		f1co 6816	. . . . . . 7 ⊢ ((𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴) → (𝑥 ∘ 𝑦):𝐴⟶1-1𝐴)
24	23	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴)) → (𝑥 ∘ 𝑦):𝐴⟶1-1𝐴)
25		f1f 6803	. . . . . . . . . . . 12 ⊢ (𝑦:𝐴⟶1-1𝐴 → 𝑦:𝐴⟶𝐴)
26	4, 25	anim12i 625	. . . . . . . . . . 11 ⊢ ((𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴) → (𝑥:𝐴⟶𝐴 ∧ 𝑦:𝐴⟶𝐴))
27	5, 6	elefmndbas 19101	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ (Base‘𝑀) ↔ 𝑦:𝐴⟶𝐴))
28	7, 27	anbi12d 644	. . . . . . . . . . 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⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))))
30	29	imp 412	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴)) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)))
31		eqid 2763	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
32	5, 6, 31	efmndov 19109	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ∘ 𝑦))
33	30, 32	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ∘ 𝑦))
34	33	eleq1d 2849	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴)) → ((𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ↔ (𝑥 ∘ 𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴}))
35	1, 19	coex 7969	. . . . . . . 8 ⊢ (𝑥 ∘ 𝑦) ∈ V
36		f1eq1 6798	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∘ 𝑦) → (ℎ:𝐴⟶1-1𝐴 ↔ (𝑥 ∘ 𝑦):𝐴⟶1-1𝐴))
37	35, 36	elab 3636	. . . . . . 7 ⊢ ((𝑥 ∘ 𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ↔ (𝑥 ∘ 𝑦):𝐴⟶1-1𝐴)
38	34, 37	bitrdi 290	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴)) → ((𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ↔ (𝑥 ∘ 𝑦):𝐴⟶1-1𝐴))
39	24, 38	mpbird 260	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴)) → (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴})
40	39	ex 418	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥:𝐴⟶1-1𝐴 ∧ 𝑦:𝐴⟶1-1𝐴) → (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴}))
41	22, 40	biimtrid 245	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ∧ 𝑦 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴}) → (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴}))
42	41	ralrimivv 3207	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴}∀𝑦 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴})
43	5	efmndmnd 19117	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
44		eqid 2763	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
45	6, 44, 31	issubm 19030	. . 3 ⊢ (𝑀 ∈ Mnd → ({ℎ ∣ ℎ:𝐴⟶1-1𝐴} ∈ (SubMnd‘𝑀) ↔ ({ℎ ∣ ℎ:𝐴⟶1-1𝐴} ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ∧ ∀𝑥 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴}∀𝑦 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴})))
46	43, 45	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → ({ℎ ∣ ℎ:𝐴⟶1-1𝐴} ∈ (SubMnd‘𝑀) ↔ ({ℎ ∣ ℎ:𝐴⟶1-1𝐴} ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ∧ ∀𝑥 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴}∀𝑦 ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴} (𝑥(+g‘𝑀)𝑦) ∈ {ℎ ∣ ℎ:𝐴⟶1-1𝐴})))
47	10, 18, 42, 46	mpbir3and 1361	1 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴⟶1-1𝐴} ∈ (SubMnd‘𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   I cid 5550   ↾ cres 5658   ∘ ccom 5660  ⟶wf 6557  ⟶1-1wf1 6558  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  +_gcplusg 17460  0_gc0g 17642  Mndcmnd 18955  SubMndcsubmnd 19009  EndoFMndcefmnd 19096

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-3 12436  df-4 12437  df-5 12438  df-6 12439  df-7 12440  df-8 12441  df-9 12442  df-n0 12637  df-z 12724  df-uz 12996  df-fz 13670  df-struct 17357  df-slot 17392  df-ndx 17404  df-base 17420  df-plusg 17473  df-tset 17479  df-0g 17644  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-submnd 19011  df-efmnd 19097

This theorem is used by:  symgsubmefmnd  19645