Theorem mhmex 12929

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 6682	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
2		basfn 12573	. . . . . 6 ⊢ Base Fn V
3		simpr 110	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ Mnd)
4	3	elexd 2765	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ V)
5		funfvex 5551	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑇 ∈ dom Base) → (Base‘𝑇) ∈ V)
6	5	funfni 5335	. . . . . 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 2765	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ V)
10		funfvex 5551	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
11	10	funfni 5335	. . . . . 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 5930	. . . . 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 4161	. . . 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 5534	. . . . . 6 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
18	17	oveq2d 5913	. . . . 5 ⊢ (𝑠 = 𝑆 → ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) = ((Base‘𝑡) ↑𝑚 (Base‘𝑆)))
19		fveq2 5534	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
20	19	oveqd 5914	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥(+g‘𝑆)𝑦))
21	20	fveqeq2d 5542	. . . . . . . 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 2698	. . . . . . 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 2698	. . . . . 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 5534	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = (0g‘𝑆))
25	24	fveqeq2d 5542	. . . . . 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 2747	. . . 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 5534	. . . . . 6 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
29	28	oveq1d 5912	. . . . 5 ⊢ (𝑡 = 𝑇 → ((Base‘𝑡) ↑𝑚 (Base‘𝑆)) = ((Base‘𝑇) ↑𝑚 (Base‘𝑆)))
30		fveq2 5534	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
31	30	oveqd 5914	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)))
32	31	eqeq2d 2201	. . . . . . 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 2514	. . . . . 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 5534	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = (0g‘𝑇))
35	34	eqeq2d 2201	. . . . . 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 2747	. . . 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 12926	. . . 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 6032	. . 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 2266	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  ∀wral 2468  {crab 2472  Vcvv 2752   × cxp 4642   Fn wfn 5230  ‘cfv 5235  (class class class)co 5897   ↑_𝑚 cmap 6675  Basecbs 12515  +_gcplusg 12592  0_gc0g 12764  Mndcmnd 12892   MndHom cmhm 12924

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-map 6677  df-inn 8951  df-ndx 12518  df-slot 12519  df-base 12521  df-mhm 12926

This theorem is referenced by:  ghmex  13211