Theorem ismhm 13607

Description: Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)

Hypotheses

Ref	Expression
ismhm.b	⊢ 𝐵 = (Base‘𝑆)
ismhm.c	⊢ 𝐶 = (Base‘𝑇)
ismhm.p	⊢ + = (+g‘𝑆)
ismhm.q	⊢ ⨣ = (+g‘𝑇)
ismhm.z	⊢ 0 = (0g‘𝑆)
ismhm.y	⊢ 𝑌 = (0g‘𝑇)

Assertion

Ref	Expression
ismhm	⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   + (𝑥,𝑦)   ⨣ (𝑥,𝑦)   𝑌(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem ismhm

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

Step	Hyp	Ref	Expression
1		df-mhm 13605	. . 3 ⊢ 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))})
2	1	elmpocl 6227	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
3		fnmap 6867	. . . . . . 7 ⊢ ↑𝑚 Fn (V × V)
4		ismhm.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑇)
5		basfn 13204	. . . . . . . . 9 ⊢ Base Fn V
6		simpr 110	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ Mnd)
7	6	elexd 2817	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ V)
8		funfvex 5665	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑇 ∈ dom Base) → (Base‘𝑇) ∈ V)
9	8	funfni 5439	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑇 ∈ V) → (Base‘𝑇) ∈ V)
10	5, 7, 9	sylancr 414	. . . . . . . 8 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (Base‘𝑇) ∈ V)
11	4, 10	eqeltrid 2318	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐶 ∈ V)
12		ismhm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
13		simpl 109	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ Mnd)
14	13	elexd 2817	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ V)
15		funfvex 5665	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
16	15	funfni 5439	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
17	5, 14, 16	sylancr 414	. . . . . . . 8 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (Base‘𝑆) ∈ V)
18	12, 17	eqeltrid 2318	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐵 ∈ V)
19		fnovex 6061	. . . . . . 7 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐶 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ↑𝑚 𝐵) ∈ V)
20	3, 11, 18, 19	mp3an2i 1379	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐶 ↑𝑚 𝐵) ∈ V)
21		rabexg 4238	. . . . . 6 ⊢ ((𝐶 ↑𝑚 𝐵) ∈ V → {𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)} ∈ V)
22	20, 21	syl 14	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → {𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)} ∈ V)
23		fveq2 5648	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
24	23, 4	eqtr4di 2282	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
25		fveq2 5648	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
26	25, 12	eqtr4di 2282	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
27	24, 26	oveqan12rd 6048	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) = (𝐶 ↑𝑚 𝐵))
28	26	adantr 276	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
29		fveq2 5648	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
30		ismhm.p	. . . . . . . . . . . . . 14 ⊢ + = (+g‘𝑆)
31	29, 30	eqtr4di 2282	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = + )
32	31	oveqd 6045	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥 + 𝑦))
33	32	fveq2d 5652	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑓‘(𝑥(+g‘𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
34		fveq2 5648	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
35		ismhm.q	. . . . . . . . . . . . 13 ⊢ ⨣ = (+g‘𝑇)
36	34, 35	eqtr4di 2282	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = ⨣ )
37	36	oveqd 6045	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)))
38	33, 37	eqeqan12d 2247	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))))
39	28, 38	raleqbidv 2747	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))))
40	28, 39	raleqbidv 2747	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))))
41		fveq2 5648	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = (0g‘𝑆))
42		ismhm.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑆)
43	41, 42	eqtr4di 2282	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = 0 )
44	43	fveq2d 5652	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑓‘(0g‘𝑠)) = (𝑓‘ 0 ))
45		fveq2 5648	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = (0g‘𝑇))
46		ismhm.y	. . . . . . . . . 10 ⊢ 𝑌 = (0g‘𝑇)
47	45, 46	eqtr4di 2282	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = 𝑌)
48	44, 47	eqeqan12d 2247	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑓‘(0g‘𝑠)) = (0g‘𝑡) ↔ (𝑓‘ 0 ) = 𝑌))
49	40, 48	anbi12d 473	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 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 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)))
50	27, 49	rabeqbidv 2798	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 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 ∈ (𝐶 ↑𝑚 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)})
51	50, 1	ovmpoga 6161	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ {𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)} ∈ V) → (𝑆 MndHom 𝑇) = {𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)})
52	22, 51	mpd3an3 1375	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) = {𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)})
53	52	eleq2d 2301	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)}))
54	11, 18	elmapd 6874	. . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝐶 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐶))
55	54	anbi1d 465	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)) ↔ (𝐹:𝐵⟶𝐶 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌))))
56		fveq1 5647	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
57		fveq1 5647	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
58		fveq1 5647	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
59	57, 58	oveq12d 6046	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
60	56, 59	eqeq12d 2246	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
61	60	2ralbidv 2557	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
62		fveq1 5647	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘ 0 ) = (𝐹‘ 0 ))
63	62	eqeq1d 2240	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘ 0 ) = 𝑌 ↔ (𝐹‘ 0 ) = 𝑌))
64	61, 63	anbi12d 473	. . . . 5 ⊢ (𝑓 = 𝐹 → ((∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌) ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
65	64	elrab 2963	. . . 4 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)} ↔ (𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
66		3anass 1009	. . . 4 ⊢ ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌) ↔ (𝐹:𝐵⟶𝐶 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
67	55, 65, 66	3bitr4g 223	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ {𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)} ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
68	53, 67	bitrd 188	. 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
69	2, 68	biadanii 617	1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511  {crab 2515  Vcvv 2803   × cxp 4729   Fn wfn 5328  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028   ↑_𝑚 cmap 6860  Basecbs 13145  +_gcplusg 13223  0_gc0g 13402  Mndcmnd 13562   MndHom cmhm 13603

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151  df-mhm 13605

This theorem is referenced by:  mhmf  13611  mhmpropd  13612  mhmlin  13613  mhm0  13614  idmhm  13615  mhmf1o  13616  0mhm  13632  resmhm  13633  resmhm2  13634  resmhm2b  13635  mhmco  13636  mhmfmhm  13767  ghmmhm  13903  srglmhm  14070  srgrmhm  14071  dfrhm2  14232  isrhm2d  14243