Theorem mgmhmeql 18742

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 2735	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2735	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	mgmhmf 18723	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4	3	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5	4	ffnd 6738	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
6	1, 2	mgmhmf 18723	. . . . 5 ⊢ (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
7	6	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
8	7	ffnd 6738	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
9		fndmin 7065	. . 3 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
10	5, 8, 9	syl2anc 584	. 2 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
11		ssrab2 4090	. . . 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 18720	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
14	13	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
15	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
16	15	ad2antrr 726	. . . . . . . . . . 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 777	. . . . . . . . . . 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 771	. . . . . . . . . . 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 2735	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
20	1, 19	mgmcl 18669	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
21	16, 17, 18, 20	syl3anc 1370	. . . . . . . . . 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 778	. . . . . . . . . . . 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 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‘𝑦) = (𝐺‘𝑦))
24	22, 23	oveq12d 7449	. . . . . . . . . . 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 775	. . . . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (+g‘𝑇) = (+g‘𝑇)
27	1, 19, 26	mgmhmlin 18725	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 776	. . . . . . . . . . . 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 18725	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 2785	. . . . . . . . . 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 6907	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → (𝐹‘𝑧) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
34		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → (𝐺‘𝑧) = (𝐺‘(𝑥(+g‘𝑆)𝑦)))
35	33, 34	eqeq12d 2751	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘(𝑥(+g‘𝑆)𝑦)) = (𝐺‘(𝑥(+g‘𝑆)𝑦))))
36	35	elrab 3695	. . . . . . . . . 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 583	. . . . . . . . 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 456	. . . . . . . 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 3144	. . . . . . 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 6907	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
41		fveq2 6907	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝐺‘𝑧) = (𝐺‘𝑦))
42	40, 41	eqeq12d 2751	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑦) = (𝐺‘𝑦)))
43	42	ralrab 3702	. . . . . . 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 234	. . . . . 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 456	. . . . 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 3144	. . . 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 6907	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
48		fveq2 6907	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥))
49	47, 48	eqeq12d 2751	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
50	49	ralrab 3702	. . . 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 234	. . 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 18728	. . . 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 713	. 2 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMgm‘𝑆))
55	10, 54	eqeltrd 2839	1 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMgm‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433   ∩ cin 3962   ⊆ wss 3963  dom cdm 5689   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  Mgmcmgm 18664   MgmHom cmgmhm 18716  SubMgmcsubmgm 18717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-mgm 18666  df-mgmhm 18718  df-submgm 18719

This theorem is referenced by: (None)