Theorem basqtop 24074

Description: An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.)

Hypothesis

Ref	Expression
qtopcmp.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
basqtop	⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) → (𝐽 qTop 𝐹) ∈ TopBases)

Proof of Theorem basqtop

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

Step	Hyp	Ref	Expression
1		f1ofo 6860	. . . . 5 ⊢ (𝐹:𝑋⟶1-1onto𝑌 → 𝐹:𝑋⟶onto𝑌)
2		qtopcmp.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	elqtop2 24064	. . . . . 6 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶onto𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
4	2	elqtop2 24064	. . . . . 6 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))
5	3, 4	anbi12d 644	. . . . 5 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶onto𝑌) → ((𝑥 ∈ (𝐽 qTop 𝐹) ∧ 𝑦 ∈ (𝐽 qTop 𝐹)) ↔ ((𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽))))
6	1, 5	sylan2 605	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) → ((𝑥 ∈ (𝐽 qTop 𝐹) ∧ 𝑦 ∈ (𝐽 qTop 𝐹)) ↔ ((𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽))))
7		simpl1l 1243	. . . . . . . . 9 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝐽 ∈ TopBases)
8		simpl2r 1246	. . . . . . . . 9 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹 “ 𝑥) ∈ 𝐽)
9		simpl3r 1248	. . . . . . . . 9 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹 “ 𝑦) ∈ 𝐽)
10		simpl1r 1244	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝐹:𝑋⟶1-1onto𝑌)
11		f1ocnv 6865	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶1-1onto𝑌 → ◡𝐹:𝑌⟶1-1onto𝑋)
12		f1ofn 6853	. . . . . . . . . . . 12 ⊢ (◡𝐹:𝑌⟶1-1onto𝑋 → ◡𝐹 Fn 𝑌)
13	10, 11, 12	3syl 19	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ◡𝐹 Fn 𝑌)
14		simpl2l 1245	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑥 ⊆ 𝑌)
15		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ (𝑥 ∩ 𝑦))
16	15	elin1d 4104	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ 𝑥)
17		fnfvima 7268	. . . . . . . . . . 11 ⊢ ((◡𝐹 Fn 𝑌 ∧ 𝑥 ⊆ 𝑌 ∧ 𝑧 ∈ 𝑥) → (◡𝐹‘𝑧) ∈ (◡𝐹 “ 𝑥))
18	13, 14, 16, 17	syl3anc 1399	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹‘𝑧) ∈ (◡𝐹 “ 𝑥))
19		simpl3l 1247	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑦 ⊆ 𝑌)
20	15	elin2d 4105	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ 𝑦)
21		fnfvima 7268	. . . . . . . . . . 11 ⊢ ((◡𝐹 Fn 𝑌 ∧ 𝑦 ⊆ 𝑌 ∧ 𝑧 ∈ 𝑦) → (◡𝐹‘𝑧) ∈ (◡𝐹 “ 𝑦))
22	13, 19, 20, 21	syl3anc 1399	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹‘𝑧) ∈ (◡𝐹 “ 𝑦))
23	18, 22	elind 4100	. . . . . . . . 9 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹‘𝑧) ∈ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)))
24		basis2 23313	. . . . . . . . 9 ⊢ (((𝐽 ∈ TopBases ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ ((◡𝐹 “ 𝑦) ∈ 𝐽 ∧ (◡𝐹‘𝑧) ∈ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)))) → ∃𝑤 ∈ 𝐽 ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))
25	7, 8, 9, 23, 24	syl22anc 852	. . . . . . . 8 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐽 ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))
26	10	adantr 486	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐹:𝑋⟶1-1onto𝑌)
27		inss1 4136	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
28		simp2l 1218	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑥 ⊆ 𝑌)
29	27, 28	sstrid 3924	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ 𝑌)
30	29	sselda 3913	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ 𝑌)
31	30	adantr 486	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑧 ∈ 𝑌)
32		f1ocnvfv2 7314	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶1-1onto𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
33	26, 31, 32	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
34		f1ofn 6853	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶1-1onto𝑌 → 𝐹 Fn 𝑋)
35	26, 34	syl 18	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐹 Fn 𝑋)
36		simprrr 794	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)))
37		inss1 4136	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ 𝑥)
38	36, 37	sstrdi 3925	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ (◡𝐹 “ 𝑥))
39		cnvimass 6215	. . . . . . . . . . . . 13 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
40		f1odm 6856	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶1-1onto𝑌 → dom 𝐹 = 𝑋)
41	26, 40	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → dom 𝐹 = 𝑋)
42	39, 41	sseqtrid 3955	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹 “ 𝑥) ⊆ 𝑋)
43	38, 42	sstrd 3923	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ 𝑋)
44		simprrl 793	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹‘𝑧) ∈ 𝑤)
45		fnfvima 7268	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ⊆ 𝑋 ∧ (◡𝐹‘𝑧) ∈ 𝑤) → (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑤))
46	35, 43, 44, 45	syl3anc 1399	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑤))
47	33, 46	eqeltrrd 2865	. . . . . . . . 9 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑧 ∈ (𝐹 “ 𝑤))
48		imassrn 6214	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑤) ⊆ ran 𝐹
49	26, 1	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐹:𝑋⟶onto𝑌)
50		forn 6827	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋⟶onto𝑌 → ran 𝐹 = 𝑌)
51	49, 50	syl 18	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → ran 𝐹 = 𝑌)
52	48, 51	sseqtrid 3955	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ⊆ 𝑌)
53		f1of1 6851	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶1-1onto𝑌 → 𝐹:𝑋⟶1-1𝑌)
54	26, 53	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐹:𝑋⟶1-1𝑌)
55		f1imacnv 6869	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑋⟶1-1𝑌 ∧ 𝑤 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤)
56	54, 43, 55	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤)
57		simprl 783	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ∈ 𝐽)
58	56, 57	eqeltrd 2864	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)
59	7	adantr 486	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐽 ∈ TopBases)
60	2	elqtop2 24064	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶onto𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
61	59, 49, 60	syl2anc 596	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
62	52, 58, 61	mpbir2and 726	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹))
63		fnfun 6667	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
64		inpreima 7093	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑥 ∩ 𝑦)) = ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)))
65	35, 63, 64	3syl 19	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹 “ (𝑥 ∩ 𝑦)) = ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)))
66	36, 65	sseqtrrd 3950	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ (◡𝐹 “ (𝑥 ∩ 𝑦)))
67	35, 63	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → Fun 𝐹)
68	38, 39	sstrdi 3925	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ dom 𝐹)
69		funimass3 7082	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (◡𝐹 “ (𝑥 ∩ 𝑦))))
70	67, 68, 69	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → ((𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (◡𝐹 “ (𝑥 ∩ 𝑦))))
71	66, 70	mpbird 260	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦))
72		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
73	72	inex1 5282	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ∈ V
74	73	elpw2 5301	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑤) ∈ 𝒫 (𝑥 ∩ 𝑦) ↔ (𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦))
75	71, 74	sylibr 237	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ∈ 𝒫 (𝑥 ∩ 𝑦))
76	62, 75	elind 4100	. . . . . . . . 9 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
77		elunii 4875	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝐹 “ 𝑤) ∧ (𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
78	47, 76, 77	syl2anc 596	. . . . . . . 8 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
79	25, 78	rexlimddv 3173	. . . . . . 7 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
80	79	ex 418	. . . . . 6 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))))
81	80	ssrdv 3919	. . . . 5 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
82	81	3expib 1140	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) → (((𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))))
83	6, 82	sylbid 243	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) → ((𝑥 ∈ (𝐽 qTop 𝐹) ∧ 𝑦 ∈ (𝐽 qTop 𝐹)) → (𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))))
84	83	ralrimivv 3207	. 2 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) → ∀𝑥 ∈ (𝐽 qTop 𝐹)∀𝑦 ∈ (𝐽 qTop 𝐹)(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
85		ovex 7482	. . 3 ⊢ (𝐽 qTop 𝐹) ∈ V
86		isbasisg 23309	. . 3 ⊢ ((𝐽 qTop 𝐹) ∈ V → ((𝐽 qTop 𝐹) ∈ TopBases ↔ ∀𝑥 ∈ (𝐽 qTop 𝐹)∀𝑦 ∈ (𝐽 qTop 𝐹)(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))))
87	85, 86	ax-mp 5	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ TopBases ↔ ∀𝑥 ∈ (𝐽 qTop 𝐹)∀𝑦 ∈ (𝐽 qTop 𝐹)(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
88	84, 87	sylibr 237	1 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋⟶1-1onto𝑌) → (𝐽 qTop 𝐹) ∈ TopBases)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4870  ^◡ccnv 5655  dom cdm 5656  ran crn 5657   “ cima 5659  Fun wfun 6558   Fn wfn 6559  ⟶1-1wf1 6561  ⟶ontowfo 6562  ⟶1-1ontowf1o 6563  ‘cfv 6564  (class class class)co 7449   qTop cqtop 17716  TopBasesctb 23307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-ov 7452  df-oprab 7453  df-mpo 7454  df-qtop 17720  df-bases 23308

This theorem is used by: (None)