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Theorem basqtop 23574

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 6789	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
2		qtopcmp.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	elqtop2 23564	. . . . . 6 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
4	2	elqtop2 23564	. . . . . 6 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑦) ∈ 𝐽)))
5	3, 4	anbi12d 632	. . . . 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 593	. . . 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 1225	. . . . . . . . 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 1228	. . . . . . . . 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 1230	. . . . . . . . 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 1226	. . . . . . . . . . . 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 6794	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
12		f1ofn 6783	. . . . . . . . . . . 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 1227	. . . . . . . . . . 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 4163	. . . . . . . . . . 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 7189	. . . . . . . . . . 11 ⊢ ((^◡𝐹 Fn 𝑌 ∧ 𝑥 ⊆ 𝑌 ∧ 𝑧 ∈ 𝑥) → (^◡𝐹‘𝑧) ∈ (^◡𝐹 “ 𝑥))
18	13, 14, 16, 17	syl3anc 1373	. . . . . . . . . 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 1229	. . . . . . . . . . 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 4164	. . . . . . . . . . 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 7189	. . . . . . . . . . 11 ⊢ ((^◡𝐹 Fn 𝑌 ∧ 𝑦 ⊆ 𝑌 ∧ 𝑧 ∈ 𝑦) → (^◡𝐹‘𝑧) ∈ (^◡𝐹 “ 𝑦))
22	13, 19, 20, 21	syl3anc 1373	. . . . . . . . . 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 4159	. . . . . . . . 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 22814	. . . . . . . . 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 838	. . . . . . . 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 4196	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
28		simp2l 1200	. . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . 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 3943	. . . . . . . . . . . 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 7234	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
33	26, 31, 32	syl2anc 584	. . . . . . . . . 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 6783	. . . . . . . . . . . 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 781	. . . . . . . . . . . . 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 4196	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 “ 𝑥) ∩ (^◡𝐹 “ 𝑦)) ⊆ (^◡𝐹 “ 𝑥)
38	36, 37	sstrdi 3956	. . . . . . . . . . . 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 6042	. . . . . . . . . . . . 13 ⊢ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹
40		f1odm 6786	. . . . . . . . . . . . . 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 3986	. . . . . . . . . . . 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 3954	. . . . . . . . . . 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 780	. . . . . . . . . . 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 7189	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ⊆ 𝑋 ∧ (^◡𝐹‘𝑧) ∈ 𝑤) → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑤))
46	35, 43, 44, 45	syl3anc 1373	. . . . . . . . . 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 2829	. . . . . . . . 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 6031	. . . . . . . . . . . 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 6757	. . . . . . . . . . . . 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 3986	. . . . . . . . . . 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 6781	. . . . . . . . . . . . . 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 6798	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑤 ⊆ 𝑋) → (^◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤)
56	54, 43, 55	syl2anc 584	. . . . . . . . . . . 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 770	. . . . . . . . . . . 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 2828	. . . . . . . . . . 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 23564	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
61	59, 49, 60	syl2anc 584	. . . . . . . . . . 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 713	. . . . . . . . . 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 6600	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
64		inpreima 7018	. . . . . . . . . . . . . 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 3981	. . . . . . . . . . . 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 3956	. . . . . . . . . . . . 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 7008	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (^◡𝐹 “ (𝑥 ∩ 𝑦))))
70	67, 68, 69	syl2anc 584	. . . . . . . . . . . 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 3448	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
73	72	inex1 5267	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ∈ V
74	73	elpw2 5284	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑤) ∈ 𝒫 (𝑥 ∩ 𝑦) ↔ (𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦))
75	71, 74	sylibr 234	. . . . . . . . . 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 4159	. . . . . . . . 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 4872	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝐹 “ 𝑤) ∧ (𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))
78	47, 76, 77	syl2anc 584	. . . . . . . 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 3140	. . . . . . 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 3949	. . . . 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 1122	. . . 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 240	. . 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 3176	. 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 7402	. . 3 ⊢ (𝐽 qTop 𝐹) ∈ V
86		isbasisg 22810	. . 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 234	1 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝐽 qTop 𝐹) ∈ TopBases)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 Vcvv 3444 ∩ cin 3910 ⊆ wss 3911 𝒫 cpw 4559 ∪ cuni 4867 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 “ cima 5634 Fun wfun 6493 Fn wfn 6494 –1-1→wf1 6496 –onto→wfo 6497 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 qTop cqtop 17442 TopBasesctb 22808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-qtop 17446 df-bases 22809

This theorem is referenced by: (None)

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