Theorem mgmhmeql 18685

Description: The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.)

Assertion

Ref	Expression
mgmhmeql	⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMgm‘𝑆))

Proof of Theorem mgmhmeql

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2728	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2728	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	mgmhmf 18666	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4	3	adantr 479	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5	4	ffnd 6728	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
6	1, 2	mgmhmf 18666	. . . . 5 ⊢ (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
7	6	adantl 480	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
8	7	ffnd 6728	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
9		fndmin 7059	. . 3 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
10	5, 8, 9	syl2anc 582	. 2 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
11		ssrab2 4077	. . . 4 ⊢ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆)
12	11	a1i 11	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆))
13		mgmhmrcl 18663	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
14	13	simpld 493	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
15	14	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
16	15	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑆 ∈ Mgm)
17		simplrl 775	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑥 ∈ (Base‘𝑆))
18		simprl 769	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑦 ∈ (Base‘𝑆))
19		eqid 2728	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
20	1, 19	mgmcl 18612	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
21	16, 17, 18, 20	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
22		simplrr 776	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘𝑥) = (𝐺‘𝑥))
23		simprr 771	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘𝑦) = (𝐺‘𝑦))
24	22, 23	oveq12d 7444	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
25		simplll 773	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
26		eqid 2728	. . . . . . . . . . . . 13 ⊢ (+g‘𝑇) = (+g‘𝑇)
27	1, 19, 26	mgmhmlin 18668	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
28	25, 17, 18, 27	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
29		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝐺 ∈ (𝑆 MgmHom 𝑇))
30	1, 19, 26	mgmhmlin 18668	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
31	29, 17, 18, 30	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
32	24, 28, 31	3eqtr4d 2778	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = (𝐺‘(𝑥(+g‘𝑆)𝑦)))
33		fveq2 6902	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → (𝐹‘𝑧) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
34		fveq2 6902	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → (𝐺‘𝑧) = (𝐺‘(𝑥(+g‘𝑆)𝑦)))
35	33, 34	eqeq12d 2744	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘(𝑥(+g‘𝑆)𝑦)) = (𝐺‘(𝑥(+g‘𝑆)𝑦))))
36	35	elrab 3684	. . . . . . . . . 10 ⊢ ((𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ↔ ((𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑥(+g‘𝑆)𝑦)) = (𝐺‘(𝑥(+g‘𝑆)𝑦))))
37	21, 32, 36	sylanbrc 581	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
38	37	expr 455	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
39	38	ralrimiva 3143	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
40		fveq2 6902	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
41		fveq2 6902	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝐺‘𝑧) = (𝐺‘𝑦))
42	40, 41	eqeq12d 2744	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑦) = (𝐺‘𝑦)))
43	42	ralrab 3690	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
44	39, 43	sylibr 233	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
45	44	expr 455	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹‘𝑥) = (𝐺‘𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
46	45	ralrimiva 3143	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
47		fveq2 6902	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
48		fveq2 6902	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥))
49	47, 48	eqeq12d 2744	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
50	49	ralrab 3690	. . . 4 ⊢ (∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ↔ ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
51	46, 50	sylibr 233	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
52	1, 19	issubmgm 18671	. . . 4 ⊢ (𝑆 ∈ Mgm → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMgm‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆) ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})))
53	15, 52	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMgm‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆) ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})))
54	12, 51, 53	mpbir2and 711	. 2 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMgm‘𝑆))
55	10, 54	eqeltrd 2829	1 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMgm‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  {crab 3430   ∩ cin 3948   ⊆ wss 3949  dom cdm 5682   Fn wfn 6548  ⟶wf 6549  ‘cfv 6553  (class class class)co 7426  Basecbs 17189  +_gcplusg 17242  Mgmcmgm 18607   MgmHom cmgmhm 18659  SubMgmcsubmgm 18660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8855  df-mgm 18609  df-mgmhm 18661  df-submgm 18662

This theorem is referenced by: (None)