Theorem mhmex 13752

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 6923	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
2		basfn 13394	. . . . . 6 ⊢ Base Fn V
3		simpr 110	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ Mnd)
4	3	elexd 2835	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ V)
5		funfvex 5710	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑇 ∈ dom Base) → (Base‘𝑇) ∈ V)
6	5	funfni 5481	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑇 ∈ V) → (Base‘𝑇) ∈ V)
7	2, 4, 6	sylancr 418	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (Base‘𝑇) ∈ V)
8		simpl 109	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ Mnd)
9	8	elexd 2835	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ V)
10		funfvex 5710	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
11	10	funfni 5481	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
12	2, 9, 11	sylancr 418	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (Base‘𝑆) ∈ V)
13		fnovex 6112	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ (Base‘𝑇) ∈ V ∧ (Base‘𝑆) ∈ V) → ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∈ V)
14	1, 7, 12, 13	mp3an2i 1383	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∈ V)
15		rabexg 4277	. . . 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 5693	. . . . . 6 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
18	17	oveq2d 6095	. . . . 5 ⊢ (𝑠 = 𝑆 → ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) = ((Base‘𝑡) ↑𝑚 (Base‘𝑆)))
19		fveq2 5693	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
20	19	oveqd 6096	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥(+g‘𝑆)𝑦))
21	20	fveqeq2d 5701	. . . . . . . 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 2765	. . . . . . 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 2765	. . . . . 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 5693	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = (0g‘𝑆))
25	24	fveqeq2d 5701	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑓‘(0g‘𝑠)) = (0g‘𝑡) ↔ (𝑓‘(0g‘𝑆)) = (0g‘𝑡)))
26	23, 25	anbi12d 477	. . . . 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 2816	. . . 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 5693	. . . . . 6 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
29	28	oveq1d 6094	. . . . 5 ⊢ (𝑡 = 𝑇 → ((Base‘𝑡) ↑𝑚 (Base‘𝑆)) = ((Base‘𝑇) ↑𝑚 (Base‘𝑆)))
30		fveq2 5693	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
31	30	oveqd 6096	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)))
32	31	eqeq2d 2250	. . . . . . 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 2574	. . . . . 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 5693	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = (0g‘𝑇))
35	34	eqeq2d 2250	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑓‘(0g‘𝑆)) = (0g‘𝑡) ↔ (𝑓‘(0g‘𝑆)) = (0g‘𝑇)))
36	33, 35	anbi12d 477	. . . . 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 2816	. . . 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 13749	. . . 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 6217	. . 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 1379	. 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 2315	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   × cxp 4770   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   ↑_𝑚 cmap 6916  Basecbs 13335  +_gcplusg 13414  0_gc0g 13593  Mndcmnd 13712   MndHom cmhm 13747

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-map 6918  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-mhm 13749

This theorem is referenced by:  ghmex  14041