Theorem mhmex 13034

Description: The set of monoid homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)

Assertion

Ref	Expression
mhmex	⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) ∈ V)

Proof of Theorem mhmex

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

Step	Hyp	Ref	Expression
1		fnmap 6709	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
2		basfn 12676	. . . . . 6 ⊢ Base Fn V
3		simpr 110	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ Mnd)
4	3	elexd 2773	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ V)
5		funfvex 5571	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑇 ∈ dom Base) → (Base‘𝑇) ∈ V)
6	5	funfni 5354	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑇 ∈ V) → (Base‘𝑇) ∈ V)
7	2, 4, 6	sylancr 414	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (Base‘𝑇) ∈ V)
8		simpl 109	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ Mnd)
9	8	elexd 2773	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ V)
10		funfvex 5571	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
11	10	funfni 5354	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
12	2, 9, 11	sylancr 414	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (Base‘𝑆) ∈ V)
13		fnovex 5951	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ (Base‘𝑇) ∈ V ∧ (Base‘𝑆) ∈ V) → ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∈ V)
14	1, 7, 12, 13	mp3an2i 1353	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∈ V)
15		rabexg 4172	. . . 4 ⊢ (((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∈ V → {𝑓 ∈ ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∣ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑇))} ∈ V)
16	14, 15	syl 14	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → {𝑓 ∈ ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∣ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑇))} ∈ V)
17		fveq2 5554	. . . . . 6 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
18	17	oveq2d 5934	. . . . 5 ⊢ (𝑠 = 𝑆 → ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) = ((Base‘𝑡) ↑𝑚 (Base‘𝑆)))
19		fveq2 5554	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
20	19	oveqd 5935	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥(+g‘𝑆)𝑦))
21	20	fveqeq2d 5562	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))))
22	17, 21	raleqbidv 2706	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))))
23	17, 22	raleqbidv 2706	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))))
24		fveq2 5554	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = (0g‘𝑆))
25	24	fveqeq2d 5562	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑓‘(0g‘𝑠)) = (0g‘𝑡) ↔ (𝑓‘(0g‘𝑆)) = (0g‘𝑡)))
26	23, 25	anbi12d 473	. . . . 5 ⊢ (𝑠 = 𝑆 → ((∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑠)) = (0g‘𝑡)) ↔ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑡))))
27	18, 26	rabeqbidv 2755	. . . 4 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑠)) = (0g‘𝑡))} = {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑆)) ∣ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑡))})
28		fveq2 5554	. . . . . 6 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
29	28	oveq1d 5933	. . . . 5 ⊢ (𝑡 = 𝑇 → ((Base‘𝑡) ↑𝑚 (Base‘𝑆)) = ((Base‘𝑇) ↑𝑚 (Base‘𝑆)))
30		fveq2 5554	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
31	30	oveqd 5935	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)))
32	31	eqeq2d 2205	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ((𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦))))
33	32	2ralbidv 2518	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦))))
34		fveq2 5554	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = (0g‘𝑇))
35	34	eqeq2d 2205	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑓‘(0g‘𝑆)) = (0g‘𝑡) ↔ (𝑓‘(0g‘𝑆)) = (0g‘𝑇)))
36	33, 35	anbi12d 473	. . . . 5 ⊢ (𝑡 = 𝑇 → ((∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑡)) ↔ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑇))))
37	29, 36	rabeqbidv 2755	. . . 4 ⊢ (𝑡 = 𝑇 → {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑆)) ∣ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑡))} = {𝑓 ∈ ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∣ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑇))})
38		df-mhm 13031	. . . 4 ⊢ MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑠)) = (0g‘𝑡))})
39	27, 37, 38	ovmpog 6053	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ {𝑓 ∈ ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∣ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑇))} ∈ V) → (𝑆 MndHom 𝑇) = {𝑓 ∈ ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∣ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑇))})
40	16, 39	mpd3an3 1349	. 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) = {𝑓 ∈ ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∣ (∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑆)) = (0g‘𝑇))})
41	40, 16	eqeltrd 2270	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  {crab 2476  Vcvv 2760   × cxp 4657   Fn wfn 5249  ‘cfv 5254  (class class class)co 5918   ↑_𝑚 cmap 6702  Basecbs 12618  +_gcplusg 12695  0_gc0g 12867  Mndcmnd 12997   MndHom cmhm 13029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-mhm 13031

This theorem is referenced by:  ghmex  13325