mgcmntco - Mathbox for Thierry Arnoux

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Theorem mgcmntco 31854

Description: A Galois connection like statement, for two functions with same range. (Contributed by Thierry Arnoux, 26-Apr-2024.)

**Hypotheses**
Ref	Expression
mgcoval.1	⊢ 𝐴 = (Base‘𝑉)
mgcoval.2	⊢ 𝐵 = (Base‘𝑊)
mgcoval.3	⊢ ≤ = (le‘𝑉)
mgcoval.4	⊢ ≲ = (le‘𝑊)
mgcval.1	⊢ 𝐻 = (𝑉MGalConn𝑊)
mgcval.2	⊢ (𝜑 → 𝑉 ∈ Proset )
mgcval.3	⊢ (𝜑 → 𝑊 ∈ Proset )
mgccole.1	⊢ (𝜑 → 𝐹𝐻𝐺)
mgcmntco.1	⊢ 𝐶 = (Base‘𝑋)
mgcmntco.2	⊢ < = (le‘𝑋)
mgcmntco.3	⊢ (𝜑 → 𝑋 ∈ Proset )
mgcmntco.4	⊢ (𝜑 → 𝐾 ∈ (𝑉Monot𝑋))
mgcmntco.5	⊢ (𝜑 → 𝐿 ∈ (𝑊Monot𝑋))

**Assertion**
Ref	Expression
mgcmntco	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦 𝑥,𝑉,𝑦 𝑥,𝑊,𝑦 𝑥,𝑋,𝑦 𝑥,𝐹,𝑦 𝑥,𝐺,𝑦 𝑥, < ,𝑦 𝑥,𝐾,𝑦 𝑥,𝐿,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝐶(𝑥,𝑦) 𝐻(𝑥,𝑦) ≤ (𝑥,𝑦) ≲ (𝑥,𝑦)

**Proof of Theorem mgcmntco**
Step	Hyp	Ref	Expression
1		mgcmntco.3	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Proset )
2	1	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ Proset )
3		mgcval.2	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Proset )
4		mgcmntco.4	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑉Monot𝑋))
5		mgcoval.1	. . . . . . . 8 ⊢ 𝐴 = (Base‘𝑉)
6		mgcmntco.1	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑋)
7	5, 6	mntf 31845	. . . . . . 7 ⊢ ((𝑉 ∈ Proset ∧ 𝑋 ∈ Proset ∧ 𝐾 ∈ (𝑉Monot𝑋)) → 𝐾:𝐴⟶𝐶)
8	3, 1, 4, 7	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → 𝐾:𝐴⟶𝐶)
9	8	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → 𝐾:𝐴⟶𝐶)
10		mgcoval.2	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
11		mgcoval.3	. . . . . . . 8 ⊢ ≤ = (le‘𝑉)
12		mgcoval.4	. . . . . . . 8 ⊢ ≲ = (le‘𝑊)
13		mgcval.1	. . . . . . . 8 ⊢ 𝐻 = (𝑉MGalConn𝑊)
14		mgcval.3	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Proset )
15		mgccole.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹𝐻𝐺)
16	5, 10, 11, 12, 13, 3, 14, 15	mgcf2 31849	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
17	16	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) → 𝐺:𝐵⟶𝐴)
18	17	ffvelcdmda 7035	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) ∈ 𝐴)
19	9, 18	ffvelcdmd 7036	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → (𝐾‘(𝐺‘𝑦)) ∈ 𝐶)
20		mgcmntco.5	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (𝑊Monot𝑋))
21	10, 6	mntf 31845	. . . . . . 7 ⊢ ((𝑊 ∈ Proset ∧ 𝑋 ∈ Proset ∧ 𝐿 ∈ (𝑊Monot𝑋)) → 𝐿:𝐵⟶𝐶)
22	14, 1, 20, 21	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → 𝐿:𝐵⟶𝐶)
23	22	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → 𝐿:𝐵⟶𝐶)
24	5, 10, 11, 12, 13, 3, 14, 15	mgcf1 31848	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
25	24	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → 𝐹:𝐴⟶𝐵)
26	25, 18	ffvelcdmd 7036	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝐺‘𝑦)) ∈ 𝐵)
27	23, 26	ffvelcdmd 7036	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → (𝐿‘(𝐹‘(𝐺‘𝑦))) ∈ 𝐶)
28	22	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) → 𝐿:𝐵⟶𝐶)
29	28	ffvelcdmda 7035	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → (𝐿‘𝑦) ∈ 𝐶)
30	16	ffvelcdmda 7035	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) ∈ 𝐴)
31		fveq2 6842	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑦) → (𝐾‘𝑥) = (𝐾‘(𝐺‘𝑦)))
32		2fveq3 6847	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑦) → (𝐿‘(𝐹‘𝑥)) = (𝐿‘(𝐹‘(𝐺‘𝑦))))
33	31, 32	breq12d 5118	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑦) → ((𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)) ↔ (𝐾‘(𝐺‘𝑦)) < (𝐿‘(𝐹‘(𝐺‘𝑦)))))
34	33	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = (𝐺‘𝑦)) → ((𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)) ↔ (𝐾‘(𝐺‘𝑦)) < (𝐿‘(𝐹‘(𝐺‘𝑦)))))
35	30, 34	rspcdv 3573	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)) → (𝐾‘(𝐺‘𝑦)) < (𝐿‘(𝐹‘(𝐺‘𝑦)))))
36	35	imp 407	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) → (𝐾‘(𝐺‘𝑦)) < (𝐿‘(𝐹‘(𝐺‘𝑦))))
37	36	an32s 650	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → (𝐾‘(𝐺‘𝑦)) < (𝐿‘(𝐹‘(𝐺‘𝑦))))
38		mgcmntco.2	. . . . 5 ⊢ < = (le‘𝑋)
39	14	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → 𝑊 ∈ Proset )
40	20	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → 𝐿 ∈ (𝑊Monot𝑋))
41		simpr 485	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
42	3	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → 𝑉 ∈ Proset )
43	15	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → 𝐹𝐻𝐺)
44	5, 10, 11, 12, 13, 42, 39, 43, 41	mgccole2 31851	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝐺‘𝑦)) ≲ 𝑦)
45	10, 6, 12, 38, 39, 2, 40, 26, 41, 44	ismntd 31844	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → (𝐿‘(𝐹‘(𝐺‘𝑦))) < (𝐿‘𝑦))
46	6, 38	prstr 18189	. . . 4 ⊢ ((𝑋 ∈ Proset ∧ ((𝐾‘(𝐺‘𝑦)) ∈ 𝐶 ∧ (𝐿‘(𝐹‘(𝐺‘𝑦))) ∈ 𝐶 ∧ (𝐿‘𝑦) ∈ 𝐶) ∧ ((𝐾‘(𝐺‘𝑦)) < (𝐿‘(𝐹‘(𝐺‘𝑦))) ∧ (𝐿‘(𝐹‘(𝐺‘𝑦))) < (𝐿‘𝑦))) → (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦))
47	2, 19, 27, 29, 37, 45, 46	syl132anc 1388	. . 3 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) ∧ 𝑦 ∈ 𝐵) → (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦))
48	47	ralrimiva 3143	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥))) → ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦))
49	1	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑋 ∈ Proset )
50	8	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝐾:𝐴⟶𝐶)
51		simpr 485	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
52	50, 51	ffvelcdmd 7036	. . . 4 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐶)
53	16	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝐺:𝐵⟶𝐴)
54	24	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) → 𝐹:𝐴⟶𝐵)
55	54	ffvelcdmda 7035	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
56	53, 55	ffvelcdmd 7036	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘(𝐹‘𝑥)) ∈ 𝐴)
57	50, 56	ffvelcdmd 7036	. . . 4 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝐾‘(𝐺‘(𝐹‘𝑥))) ∈ 𝐶)
58	22	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝐿:𝐵⟶𝐶)
59	58, 55	ffvelcdmd 7036	. . . 4 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝐿‘(𝐹‘𝑥)) ∈ 𝐶)
60	3	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑉 ∈ Proset )
61	4	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝐾 ∈ (𝑉Monot𝑋))
62	14	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑊 ∈ Proset )
63	15	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝐹𝐻𝐺)
64	5, 10, 11, 12, 13, 60, 62, 63, 51	mgccole1 31850	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))
65	5, 6, 11, 38, 60, 49, 61, 51, 56, 64	ismntd 31844	. . . 4 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) < (𝐾‘(𝐺‘(𝐹‘𝑥))))
66	24	ffvelcdmda 7035	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
67		2fveq3 6847	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐾‘(𝐺‘𝑦)) = (𝐾‘(𝐺‘(𝐹‘𝑥))))
68		fveq2 6842	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐿‘𝑦) = (𝐿‘(𝐹‘𝑥)))
69	67, 68	breq12d 5118	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦) ↔ (𝐾‘(𝐺‘(𝐹‘𝑥))) < (𝐿‘(𝐹‘𝑥))))
70	69	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → ((𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦) ↔ (𝐾‘(𝐺‘(𝐹‘𝑥))) < (𝐿‘(𝐹‘𝑥))))
71	66, 70	rspcdv 3573	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦) → (𝐾‘(𝐺‘(𝐹‘𝑥))) < (𝐿‘(𝐹‘𝑥))))
72	71	imp 407	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) → (𝐾‘(𝐺‘(𝐹‘𝑥))) < (𝐿‘(𝐹‘𝑥)))
73	72	an32s 650	. . . 4 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝐾‘(𝐺‘(𝐹‘𝑥))) < (𝐿‘(𝐹‘𝑥)))
74	6, 38	prstr 18189	. . . 4 ⊢ ((𝑋 ∈ Proset ∧ ((𝐾‘𝑥) ∈ 𝐶 ∧ (𝐾‘(𝐺‘(𝐹‘𝑥))) ∈ 𝐶 ∧ (𝐿‘(𝐹‘𝑥)) ∈ 𝐶) ∧ ((𝐾‘𝑥) < (𝐾‘(𝐺‘(𝐹‘𝑥))) ∧ (𝐾‘(𝐺‘(𝐹‘𝑥))) < (𝐿‘(𝐹‘𝑥)))) → (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)))
75	49, 52, 57, 59, 65, 73, 74	syl132anc 1388	. . 3 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)))
76	75	ralrimiva 3143	. 2 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)) → ∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)))
77	48, 76	impbida 799	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 class class class wbr 5105 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 lecple 17140 Proset cproset 18182 Monotcmnt 31838 MGalConncmgc 31839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-map 8767 df-proset 18184 df-mnt 31840 df-mgc 31841

This theorem is referenced by: (None)

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