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Theorem basqtop 23539

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–1-1-onto→𝑌) → (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 6831	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
2		qtopcmp.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	elqtop2 23529	. . . . . 6 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
4	2	elqtop2 23529	. . . . . 6 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)))
5	3, 4	anbi12d 630	. . . . 5 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → ((𝑥 ∈ (𝐽 qTop 𝐹) ∧ 𝑦 ∈ (𝐽 qTop 𝐹)) ↔ ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽))))
6	1, 5	sylan2 592	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝑥 ∈ (𝐽 qTop 𝐹) ∧ 𝑦 ∈ (𝐽 qTop 𝐹)) ↔ ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽))))
7		simpl1l 1221	. . . . . . . . 9 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝐽 ∈ TopBases)
8		simpl2r 1224	. . . . . . . . 9 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (^◡𝐹 “ 𝑥) ∈ 𝐽)
9		simpl3r 1226	. . . . . . . . 9 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (^◡𝐹 “ 𝑦) ∈ 𝐽)
10		simpl1r 1222	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝐹:𝑋–1-1-onto→𝑌)
11		f1ocnv 6836	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
12		f1ofn 6825	. . . . . . . . . . . 12 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹 Fn 𝑌)
13	10, 11, 12	3syl 18	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ^◡𝐹 Fn 𝑌)
14		simpl2l 1223	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑥 ⊆ 𝑌)
15		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ (𝑥 ∩ 𝑦))
16	15	elin1d 4191	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ 𝑥)
17		fnfvima 7227	. . . . . . . . . . 11 ⊢ ((^◡𝐹 Fn 𝑌 ∧ 𝑥 ⊆ 𝑌 ∧ 𝑧 ∈ 𝑥) → (^◡𝐹‘𝑧) ∈ (^◡𝐹 “ 𝑥))
18	13, 14, 16, 17	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (^◡𝐹‘𝑧) ∈ (^◡𝐹 “ 𝑥))
19		simpl3l 1225	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑦 ⊆ 𝑌)
20	15	elin2d 4192	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ 𝑦)
21		fnfvima 7227	. . . . . . . . . . 11 ⊢ ((^◡𝐹 Fn 𝑌 ∧ 𝑦 ⊆ 𝑌 ∧ 𝑧 ∈ 𝑦) → (^◡𝐹‘𝑧) ∈ (^◡𝐹 “ 𝑦))
22	13, 19, 20, 21	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (^◡𝐹‘𝑧) ∈ (^◡𝐹 “ 𝑦))
23	18, 22	elind 4187	. . . . . . . . 9 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (^◡𝐹‘𝑧) ∈ ((^◡𝐹 “ 𝑥) ∩ (^◡𝐹 “ 𝑦)))
24		basis2 22778	. . . . . . . . 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 836	. . . . . . . 8 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 480	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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–1-1-onto→𝑌)
27		inss1 4221	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
28		simp2l 1196	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑥 ⊆ 𝑌)
29	27, 28	sstrid 3986	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ 𝑌)
30	29	sselda 3975	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ 𝑌)
31	30	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 7268	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
33	26, 31, 32	syl2anc 583	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 6825	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 Fn 𝑋)
35	26, 34	syl 17	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 779	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 4221	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 “ 𝑥) ∩ (^◡𝐹 “ 𝑦)) ⊆ (^◡𝐹 “ 𝑥)
38	36, 37	sstrdi 3987	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 6071	. . . . . . . . . . . . 13 ⊢ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹
40		f1odm 6828	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → dom 𝐹 = 𝑋)
41	26, 40	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 4027	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 3985	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 778	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 7227	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ⊆ 𝑋 ∧ (^◡𝐹‘𝑧) ∈ 𝑤) → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑤))
46	35, 43, 44, 45	syl3anc 1368	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 2826	. . . . . . . . 9 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 6061	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑤) ⊆ ran 𝐹
49	26, 1	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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–onto→𝑌)
50		forn 6799	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
51	49, 50	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 4027	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 6823	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
54	26, 53	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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–1-1→𝑌)
55		f1imacnv 6840	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑤 ⊆ 𝑋) → (^◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤)
56	54, 43, 55	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 768	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 2825	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 480	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 23529	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
61	59, 49, 60	syl2anc 583	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 710	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 6640	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
64		inpreima 7056	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (^◡𝐹 “ (𝑥 ∩ 𝑦)) = ((^◡𝐹 “ 𝑥) ∩ (^◡𝐹 “ 𝑦)))
65	35, 63, 64	3syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 4016	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 17	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 3987	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 7046	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (^◡𝐹 “ (𝑥 ∩ 𝑦))))
70	67, 68, 69	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 257	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 3470	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
73	72	inex1 5308	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ∈ V
74	73	elpw2 5336	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑤) ∈ 𝒫 (𝑥 ∩ 𝑦) ↔ (𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦))
75	71, 74	sylibr 233	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 4187	. . . . . . . . 9 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 4905	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝐹 “ 𝑤) ∧ (𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
78	47, 76, 77	syl2anc 583	. . . . . . . 8 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (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 3153	. . . . . . 7 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
80	79	ex 412	. . . . . 6 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))))
81	80	ssrdv 3981	. . . . 5 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
82	81	3expib 1119	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))))
83	6, 82	sylbid 239	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝑥 ∈ (𝐽 qTop 𝐹) ∧ 𝑦 ∈ (𝐽 qTop 𝐹)) → (𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))))
84	83	ralrimivv 3190	. 2 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∀𝑥 ∈ (𝐽 qTop 𝐹)∀𝑦 ∈ (𝐽 qTop 𝐹)(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
85		ovex 7435	. . 3 ⊢ (𝐽 qTop 𝐹) ∈ V
86		isbasisg 22774	. . 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 233	1 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝐽 qTop 𝐹) ∈ TopBases)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 Vcvv 3466 ∩ cin 3940 ⊆ wss 3941 𝒫 cpw 4595 ∪ cuni 4900 ^◡ccnv 5666 dom cdm 5667 ran crn 5668 “ cima 5670 Fun wfun 6528 Fn wfn 6529 –1-1→wf1 6531 –onto→wfo 6532 –1-1-onto→wf1o 6533 ‘cfv 6534 (class class class)co 7402 qTop cqtop 17450 TopBasesctb 22772

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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-qtop 17454 df-bases 22773

This theorem is referenced by: (None)

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