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Theorem tgqtop 22863

Description: An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.)

**Hypothesis**
Ref	Expression
qtopcmp.1	⊢ 𝑋 = ∪ 𝐽

**Assertion**
Ref	Expression
tgqtop	⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((topGen‘𝐽) qTop 𝐹) = (topGen‘(𝐽 qTop 𝐹)))

Proof of Theorem tgqtop Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1ocnv 6728	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
2		f1ofun 6718	. . . . . . . . 9 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → Fun ^◡𝐹)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → Fun ^◡𝐹)
4	3	ad2antlr 724	. . . . . . 7 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → Fun ^◡𝐹)
5		simpr 485	. . . . . . . 8 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝑥 ⊆ 𝑌)
6		df-rn 5600	. . . . . . . . 9 ⊢ ran 𝐹 = dom ^◡𝐹
7		f1ofo 6723	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
8	7	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝐹:𝑋–onto→𝑌)
9		forn 6691	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ran 𝐹 = 𝑌)
11	6, 10	eqtr3id 2792	. . . . . . . 8 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → dom ^◡𝐹 = 𝑌)
12	5, 11	sseqtrrd 3962	. . . . . . 7 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝑥 ⊆ dom ^◡𝐹)
13		funimass4 6834	. . . . . . 7 ⊢ ((Fun ^◡𝐹 ∧ 𝑥 ⊆ dom ^◡𝐹) → ((^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ↔ ∀𝑦 ∈ 𝑥 (^◡𝐹‘𝑦) ∈ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
14	4, 12, 13	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ↔ ∀𝑦 ∈ 𝑥 (^◡𝐹‘𝑦) ∈ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
15		dfss3 3909	. . . . . . 7 ⊢ (𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥))
16		simprl 768	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → 𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥))
17	16	elin1d 4132	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → 𝑧 ∈ (𝐽 qTop 𝐹))
18		qtopcmp.1	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 = ∪ 𝐽
19	18	elqtop2 22852	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑧 ∈ (𝐽 qTop 𝐹) ↔ (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽)))
20	7, 19	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑧 ∈ (𝐽 qTop 𝐹) ↔ (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽)))
21	20	ad3antrrr 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → (𝑧 ∈ (𝐽 qTop 𝐹) ↔ (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽)))
22	17, 21	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽))
23	22	simprd 496	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → (^◡𝐹 “ 𝑧) ∈ 𝐽)
24	16	elin2d 4133	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → 𝑧 ∈ 𝒫 𝑥)
25	24	elpwid 4544	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → 𝑧 ⊆ 𝑥)
26		imass2 6010	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑥 → (^◡𝐹 “ 𝑧) ⊆ (^◡𝐹 “ 𝑥))
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → (^◡𝐹 “ 𝑧) ⊆ (^◡𝐹 “ 𝑥))
28	23, 27	elpwd 4541	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → (^◡𝐹 “ 𝑧) ∈ 𝒫 (^◡𝐹 “ 𝑥))
29	23, 28	elind 4128	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → (^◡𝐹 “ 𝑧) ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)))
30		simp-4r 781	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → 𝐹:𝑋–1-1-onto→𝑌)
31	30, 1	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → ^◡𝐹:𝑌–1-1-onto→𝑋)
32		f1ofn 6717	. . . . . . . . . . . . . 14 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹 Fn 𝑌)
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → ^◡𝐹 Fn 𝑌)
34	5	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → 𝑥 ⊆ 𝑌)
35	25, 34	sstrd 3931	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → 𝑧 ⊆ 𝑌)
36		simprr 770	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → 𝑦 ∈ 𝑧)
37		fnfvima 7109	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 Fn 𝑌 ∧ 𝑧 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑧) → (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧))
38	33, 35, 36, 37	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧))
39		eleq2 2827	. . . . . . . . . . . . 13 ⊢ (𝑤 = (^◡𝐹 “ 𝑧) → ((^◡𝐹‘𝑦) ∈ 𝑤 ↔ (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧)))
40	39	rspcev 3561	. . . . . . . . . . . 12 ⊢ (((^◡𝐹 “ 𝑧) ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧)) → ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤)
41	29, 38, 40	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ 𝑧)) → ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤)
42	41	rexlimdvaa 3214	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → (∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧 → ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤))
43		simp-4r 781	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝐹:𝑋–1-1-onto→𝑌)
44		f1ofun 6718	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → Fun 𝐹)
45	43, 44	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → Fun 𝐹)
46		simprl 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)))
47	46	elin2d 4133	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝑤 ∈ 𝒫 (^◡𝐹 “ 𝑥))
48	47	elpwid 4544	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝑤 ⊆ (^◡𝐹 “ 𝑥))
49		funimass2 6517	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ (^◡𝐹 “ 𝑥)) → (𝐹 “ 𝑤) ⊆ 𝑥)
50	45, 48, 49	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (𝐹 “ 𝑤) ⊆ 𝑥)
51	5	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝑥 ⊆ 𝑌)
52	50, 51	sstrd 3931	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (𝐹 “ 𝑤) ⊆ 𝑌)
53		f1of1 6715	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
54	43, 53	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝐹:𝑋–1-1→𝑌)
55	46	elin1d 4132	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝑤 ∈ 𝐽)
56		elssuni 4871	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
57	56, 18	sseqtrrdi 3972	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ 𝑋)
58	55, 57	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝑤 ⊆ 𝑋)
59		f1imacnv 6732	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑤 ⊆ 𝑋) → (^◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤)
60	54, 58, 59	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (^◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤)
61	60, 55	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)
62	18	elqtop2 22852	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
63	7, 62	sylan2 593	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
64	63	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
65	52, 61, 64	mpbir2and 710	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹))
66		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
67	66	elpw2 5269	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑤) ∈ 𝒫 𝑥 ↔ (𝐹 “ 𝑤) ⊆ 𝑥)
68	50, 67	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (𝐹 “ 𝑤) ∈ 𝒫 𝑥)
69	65, 68	elind 4128	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥))
70	5	sselda 3921	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
71	70	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝑦 ∈ 𝑌)
72		f1ocnvfv2 7149	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑦)) = 𝑦)
73	43, 71, 72	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (𝐹‘(^◡𝐹‘𝑦)) = 𝑦)
74		f1ofn 6717	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 Fn 𝑋)
75	74	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐹 Fn 𝑋)
76	75	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝐹 Fn 𝑋)
77		simprr 770	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (^◡𝐹‘𝑦) ∈ 𝑤)
78		fnfvima 7109	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ⊆ 𝑋 ∧ (^◡𝐹‘𝑦) ∈ 𝑤) → (𝐹‘(^◡𝐹‘𝑦)) ∈ (𝐹 “ 𝑤))
79	76, 58, 77, 78	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → (𝐹‘(^◡𝐹‘𝑦)) ∈ (𝐹 “ 𝑤))
80	73, 79	eqeltrrd 2840	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → 𝑦 ∈ (𝐹 “ 𝑤))
81		eleq2 2827	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹 “ 𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝐹 “ 𝑤)))
82	81	rspcev 3561	. . . . . . . . . . . 12 ⊢ (((𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ (𝐹 “ 𝑤)) → ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧)
83	69, 80, 82	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) ∧ (𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ 𝑤)) → ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧)
84	83	rexlimdvaa 3214	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → (∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤 → ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧))
85	42, 84	impbid 211	. . . . . . . . 9 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → (∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧 ↔ ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤))
86		eluni2 4843	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧)
87		eluni2 4843	. . . . . . . . 9 ⊢ ((^◡𝐹‘𝑦) ∈ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ↔ ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤)
88	85, 86, 87	3bitr4g 314	. . . . . . . 8 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ (^◡𝐹‘𝑦) ∈ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
89	88	ralbidva 3111	. . . . . . 7 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 (^◡𝐹‘𝑦) ∈ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
90	15, 89	bitrid 282	. . . . . 6 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → (𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 (^◡𝐹‘𝑦) ∈ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
91	14, 90	bitr4d 281	. . . . 5 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ↔ 𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)))
92		eltg 22107	. . . . . 6 ⊢ (𝐽 ∈ TopBases → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ (^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
93	92	ad2antrr 723	. . . . 5 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ (^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
94		ovex 7308	. . . . . 6 ⊢ (𝐽 qTop 𝐹) ∈ V
95		eltg 22107	. . . . . 6 ⊢ ((𝐽 qTop 𝐹) ∈ V → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) ↔ 𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)))
96	94, 95	mp1i 13	. . . . 5 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) ↔ 𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)))
97	91, 93, 96	3bitr4d 311	. . . 4 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹))))
98	97	pm5.32da 579	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)))))
99		tgtopon 22121	. . . . . 6 ⊢ (𝐽 ∈ TopBases → (topGen‘𝐽) ∈ (TopOn‘∪ 𝐽))
100	99	adantr 481	. . . . 5 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘𝐽) ∈ (TopOn‘∪ 𝐽))
101	18	fveq2i 6777	. . . . 5 ⊢ (TopOn‘𝑋) = (TopOn‘∪ 𝐽)
102	100, 101	eleqtrrdi 2850	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘𝐽) ∈ (TopOn‘𝑋))
103	7	adantl 482	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐹:𝑋–onto→𝑌)
104		elqtop3 22854	. . . 4 ⊢ (((topGen‘𝐽) ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ ((topGen‘𝐽) qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽))))
105	102, 103, 104	syl2anc 584	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ ((topGen‘𝐽) qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽))))
106		unitg 22117	. . . . . . . . 9 ⊢ ((𝐽 qTop 𝐹) ∈ V → ∪ (topGen‘(𝐽 qTop 𝐹)) = ∪ (𝐽 qTop 𝐹))
107	94, 106	ax-mp 5	. . . . . . . 8 ⊢ ∪ (topGen‘(𝐽 qTop 𝐹)) = ∪ (𝐽 qTop 𝐹)
108	18	elqtop2 22852	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
109	7, 108	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
110		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) → 𝑥 ⊆ 𝑌)
111		velpw 4538	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝑌 ↔ 𝑥 ⊆ 𝑌)
112	110, 111	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) → 𝑥 ∈ 𝒫 𝑌)
113	109, 112	syl6bi 252	. . . . . . . . . 10 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) → 𝑥 ∈ 𝒫 𝑌))
114	113	ssrdv 3927	. . . . . . . . 9 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝐽 qTop 𝐹) ⊆ 𝒫 𝑌)
115		sspwuni 5029	. . . . . . . . 9 ⊢ ((𝐽 qTop 𝐹) ⊆ 𝒫 𝑌 ↔ ∪ (𝐽 qTop 𝐹) ⊆ 𝑌)
116	114, 115	sylib 217	. . . . . . . 8 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∪ (𝐽 qTop 𝐹) ⊆ 𝑌)
117	107, 116	eqsstrid 3969	. . . . . . 7 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∪ (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝑌)
118		sspwuni 5029	. . . . . . 7 ⊢ ((topGen‘(𝐽 qTop 𝐹)) ⊆ 𝒫 𝑌 ↔ ∪ (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝑌)
119	117, 118	sylibr 233	. . . . . 6 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝒫 𝑌)
120	119	sseld 3920	. . . . 5 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) → 𝑥 ∈ 𝒫 𝑌))
121	120, 111	syl6ib 250	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) → 𝑥 ⊆ 𝑌))
122	121	pm4.71rd 563	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)))))
123	98, 105, 122	3bitr4d 311	. 2 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ ((topGen‘𝐽) qTop 𝐹) ↔ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹))))
124	123	eqrdv 2736	1 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((topGen‘𝐽) qTop 𝐹) = (topGen‘(𝐽 qTop 𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 𝒫 cpw 4533 ∪ cuni 4839 ^◡ccnv 5588 dom cdm 5589 ran crn 5590 “ cima 5592 Fun wfun 6427 Fn wfn 6428 –1-1→wf1 6430 –onto→wfo 6431 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 topGenctg 17148 qTop cqtop 17214 TopOnctopon 22059 TopBasesctb 22095

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-topgen 17154 df-qtop 17218 df-top 22043 df-topon 22060 df-bases 22096

This theorem is referenced by: imasf1oxms 23645

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