Theorem mhmex 13550

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 6824	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
2		basfn 13146	. . . . . 6 ⊢ Base Fn V
3		simpr 110	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ Mnd)
4	3	elexd 2816	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ V)
5		funfvex 5656	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑇 ∈ dom Base) → (Base‘𝑇) ∈ V)
6	5	funfni 5432	. . . . . 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 2816	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ V)
10		funfvex 5656	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
11	10	funfni 5432	. . . . . 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 6051	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ (Base‘𝑇) ∈ V ∧ (Base‘𝑆) ∈ V) → ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∈ V)
14	1, 7, 12, 13	mp3an2i 1378	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((Base‘𝑇) ↑𝑚 (Base‘𝑆)) ∈ V)
15		rabexg 4233	. . . 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 5639	. . . . . 6 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
18	17	oveq2d 6034	. . . . 5 ⊢ (𝑠 = 𝑆 → ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) = ((Base‘𝑡) ↑𝑚 (Base‘𝑆)))
19		fveq2 5639	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
20	19	oveqd 6035	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥(+g‘𝑆)𝑦))
21	20	fveqeq2d 5647	. . . . . . . 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 2746	. . . . . . 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 2746	. . . . . 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 5639	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = (0g‘𝑆))
25	24	fveqeq2d 5647	. . . . . 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 2797	. . . 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 5639	. . . . . 6 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
29	28	oveq1d 6033	. . . . 5 ⊢ (𝑡 = 𝑇 → ((Base‘𝑡) ↑𝑚 (Base‘𝑆)) = ((Base‘𝑇) ↑𝑚 (Base‘𝑆)))
30		fveq2 5639	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
31	30	oveqd 6035	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)))
32	31	eqeq2d 2243	. . . . . . 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 2556	. . . . . 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 5639	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = (0g‘𝑇))
35	34	eqeq2d 2243	. . . . . 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 2797	. . . 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 13547	. . . 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 6156	. . 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 1374	. 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 2308	1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∀wral 2510  {crab 2514  Vcvv 2802   × cxp 4723   Fn wfn 5321  ‘cfv 5326  (class class class)co 6018   ↑_𝑚 cmap 6817  Basecbs 13087  +_gcplusg 13165  0_gc0g 13344  Mndcmnd 13504   MndHom cmhm 13545

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-map 6819  df-inn 9144  df-ndx 13090  df-slot 13091  df-base 13093  df-mhm 13547

This theorem is referenced by:  ghmex  13847