Theorem ismhm 13534

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 13532	. . 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 6212	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
3		fnmap 6819	. . . . . . 7 ⊢ ↑𝑚 Fn (V × V)
4		ismhm.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑇)
5		basfn 13131	. . . . . . . . 9 ⊢ Base Fn V
6		simpr 110	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ Mnd)
7	6	elexd 2814	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑇 ∈ V)
8		funfvex 5652	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑇 ∈ dom Base) → (Base‘𝑇) ∈ V)
9	8	funfni 5429	. . . . . . . . 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 2814	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝑆 ∈ V)
15		funfvex 5652	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
16	15	funfni 5429	. . . . . . . . 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 6046	. . . . . . 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 4231	. . . . . 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 5635	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
24	23, 4	eqtr4di 2280	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
25		fveq2 5635	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
26	25, 12	eqtr4di 2280	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
27	24, 26	oveqan12rd 6033	. . . . . . 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 5635	. . . . . . . . . . . . . 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 6030	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥 + 𝑦))
33	32	fveq2d 5639	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑓‘(𝑥(+g‘𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
34		fveq2 5635	. . . . . . . . . . . . 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 6030	. . . . . . . . . . 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 5635	. . . . . . . . . . 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 5639	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑓‘(0g‘𝑠)) = (𝑓‘ 0 ))
45		fveq2 5635	. . . . . . . . . 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 2795	. . . . . 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 6146	. . . . 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 6826	. . . . 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 5634	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
57		fveq1 5634	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
58		fveq1 5634	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
59	57, 58	oveq12d 6031	. . . . . . . 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 5634	. . . . . . 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 2960	. . . 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 2800   × cxp 4721   Fn wfn 5319  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013   ↑_𝑚 cmap 6812  Basecbs 13072  +_gcplusg 13150  0_gc0g 13329  Mndcmnd 13489   MndHom cmhm 13530

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-map 6814  df-inn 9134  df-ndx 13075  df-slot 13076  df-base 13078  df-mhm 13532

This theorem is referenced by:  mhmf  13538  mhmpropd  13539  mhmlin  13540  mhm0  13541  idmhm  13542  mhmf1o  13543  0mhm  13559  resmhm  13560  resmhm2  13561  resmhm2b  13562  mhmco  13563  mhmfmhm  13694  ghmmhm  13830  srglmhm  13996  srgrmhm  13997  dfrhm2  14158  isrhm2d  14169