Theorem ismhm 13489

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 13487	. . 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 6199	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
3		fnmap 6800	. . . . . . 7 ⊢ ↑𝑚 Fn (V × V)
4		ismhm.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑇)
5		basfn 13086	. . . . . . . . 9 ⊢ Base Fn V
6		simpr 110	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ Mnd)
7	6	elexd 2813	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ V)
8		funfvex 5643	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑇 ∈ dom Base) → (Base‘𝑇) ∈ V)
9	8	funfni 5422	. . . . . . . . 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 2316	. . . . . . 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 2813	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ V)
15		funfvex 5643	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
16	15	funfni 5422	. . . . . . . . 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 2316	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐵 ∈ V)
19		fnovex 6033	. . . . . . 7 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐶 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ↑𝑚 𝐵) ∈ V)
20	3, 11, 18, 19	mp3an2i 1376	. . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐶 ↑𝑚 𝐵) ∈ V)
21		rabexg 4226	. . . . . 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 5626	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
24	23, 4	eqtr4di 2280	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
25		fveq2 5626	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
26	25, 12	eqtr4di 2280	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
27	24, 26	oveqan12rd 6020	. . . . . . 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 5626	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
30		ismhm.p	. . . . . . . . . . . . . 14 ⊢ + = (+g‘𝑆)
31	29, 30	eqtr4di 2280	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = + )
32	31	oveqd 6017	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥 + 𝑦))
33	32	fveq2d 5630	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑓‘(𝑥(+g‘𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
34		fveq2 5626	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
35		ismhm.q	. . . . . . . . . . . . 13 ⊢ ⨣ = (+g‘𝑇)
36	34, 35	eqtr4di 2280	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = ⨣ )
37	36	oveqd 6017	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)))
38	33, 37	eqeqan12d 2245	. . . . . . . . . 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 2744	. . . . . . . . 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 2744	. . . . . . . 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 5626	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = (0g‘𝑆))
42		ismhm.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑆)
43	41, 42	eqtr4di 2280	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = 0 )
44	43	fveq2d 5630	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑓‘(0g‘𝑠)) = (𝑓‘ 0 ))
45		fveq2 5626	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = (0g‘𝑇))
46		ismhm.y	. . . . . . . . . 10 ⊢ 𝑌 = (0g‘𝑇)
47	45, 46	eqtr4di 2280	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = 𝑌)
48	44, 47	eqeqan12d 2245	. . . . . . . 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 2794	. . . . . 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 6133	. . . . 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 1372	. . . 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 2299	. . 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 6807	. . . . 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 5625	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
57		fveq1 5625	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
58		fveq1 5625	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
59	57, 58	oveq12d 6018	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
60	56, 59	eqeq12d 2244	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
61	60	2ralbidv 2554	. . . . . 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 5625	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘ 0 ) = (𝐹‘ 0 ))
63	62	eqeq1d 2238	. . . . . 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 2959	. . . 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 1006	. . . 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 615	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 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {crab 2512  Vcvv 2799   × cxp 4716   Fn wfn 5312  ⟶wf 5313  ‘cfv 5317  (class class class)co 6000   ↑_𝑚 cmap 6793  Basecbs 13027  +_gcplusg 13105  0_gc0g 13284  Mndcmnd 13444   MndHom cmhm 13485

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-map 6795  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-mhm 13487

This theorem is referenced by:  mhmf  13493  mhmpropd  13494  mhmlin  13495  mhm0  13496  idmhm  13497  mhmf1o  13498  0mhm  13514  resmhm  13515  resmhm2  13516  resmhm2b  13517  mhmco  13518  mhmfmhm  13649  ghmmhm  13785  srglmhm  13951  srgrmhm  13952  dfrhm2  14112  isrhm2d  14123