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Theorem tgqtop 22609

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 6673	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
2		f1ofun 6663	. . . . . . . . 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 727	. . . . . . 7 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → Fun ^◡𝐹)
5		simpr 488	. . . . . . . 8 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝑥 ⊆ 𝑌)
6		df-rn 5562	. . . . . . . . 9 ⊢ ran 𝐹 = dom ^◡𝐹
7		f1ofo 6668	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
8	7	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝐹:𝑋–onto→𝑌)
9		forn 6636	. . . . . . . . . 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 3942	. . . . . . 7 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝑥 ⊆ dom ^◡𝐹)
13		funimass4 6777	. . . . . . 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 587	. . . . . 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 3888	. . . . . . 7 ⊢ (𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥))
16		simprl 771	. . . . . . . . . . . . . . . 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 4112	. . . . . . . . . . . . . . 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 22598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑧 ∈ (𝐽 qTop 𝐹) ↔ (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽)))
20	7, 19	sylan2 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑧 ∈ (𝐽 qTop 𝐹) ↔ (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽)))
21	20	ad3antrrr 730	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . . 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 4113	. . . . . . . . . . . . . . . 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 4524	. . . . . . . . . . . . . . 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 5970	. . . . . . . . . . . . . . 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 4521	. . . . . . . . . . . . 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 4108	. . . . . . . . . . . 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 784	. . . . . . . . . . . . . . 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 6662	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 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 3911	. . . . . . . . . . . . 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 773	. . . . . . . . . . . . 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 7049	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 Fn 𝑌 ∧ 𝑧 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑧) → (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧))
38	33, 35, 36, 37	syl3anc 1373	. . . . . . . . . . . 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 2826	. . . . . . . . . . . . 13 ⊢ (𝑤 = (^◡𝐹 “ 𝑧) → ((^◡𝐹‘𝑦) ∈ 𝑤 ↔ (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧)))
40	39	rspcev 3537	. . . . . . . . . . . 12 ⊢ (((^◡𝐹 “ 𝑧) ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧)) → ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤)
41	29, 38, 40	syl2anc 587	. . . . . . . . . . 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 3204	. . . . . . . . . 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 784	. . . . . . . . . . . . . . . . 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 6663	. . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . 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 4113	. . . . . . . . . . . . . . . . 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 4524	. . . . . . . . . . . . . . . 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 6463	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ (^◡𝐹 “ 𝑥)) → (𝐹 “ 𝑤) ⊆ 𝑥)
50	45, 48, 49	syl2anc 587	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 3911	. . . . . . . . . . . . . 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 6660	. . . . . . . . . . . . . . . . 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 4112	. . . . . . . . . . . . . . . . 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 4851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
57	56, 18	sseqtrrdi 3952	. . . . . . . . . . . . . . . . 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 6677	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑤 ⊆ 𝑋) → (^◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤)
60	54, 58, 59	syl2anc 587	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . 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 22598	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
63	7, 62	sylan2 596	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
64	63	ad3antrrr 730	. . . . . . . . . . . . . 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 713	. . . . . . . . . . . . 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 3412	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
67	66	elpw2 5238	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑤) ∈ 𝒫 𝑥 ↔ (𝐹 “ 𝑤) ⊆ 𝑥)
68	50, 67	sylibr 237	. . . . . . . . . . . . 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 4108	. . . . . . . . . . . 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 3901	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
71	70	adantr 484	. . . . . . . . . . . . . 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 7088	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑦)) = 𝑦)
73	43, 71, 72	syl2anc 587	. . . . . . . . . . . . 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 6662	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 Fn 𝑋)
75	74	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐹 Fn 𝑋)
76	75	ad3antrrr 730	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 7049	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ⊆ 𝑋 ∧ (^◡𝐹‘𝑦) ∈ 𝑤) → (𝐹‘(^◡𝐹‘𝑦)) ∈ (𝐹 “ 𝑤))
79	76, 58, 77, 78	syl3anc 1373	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 2826	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹 “ 𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝐹 “ 𝑤)))
82	81	rspcev 3537	. . . . . . . . . . . 12 ⊢ (((𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ (𝐹 “ 𝑤)) → ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧)
83	69, 80, 82	syl2anc 587	. . . . . . . . . . 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 3204	. . . . . . . . . 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 215	. . . . . . . . 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 4823	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧)
87		eluni2 4823	. . . . . . . . 9 ⊢ ((^◡𝐹‘𝑦) ∈ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ↔ ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤)
88	85, 86, 87	3bitr4g 317	. . . . . . . 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 3117	. . . . . . 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	syl5bb 286	. . . . . 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 285	. . . . 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 21854	. . . . . 6 ⊢ (𝐽 ∈ TopBases → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ (^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
93	92	ad2antrr 726	. . . . 5 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ (^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
94		ovex 7246	. . . . . 6 ⊢ (𝐽 qTop 𝐹) ∈ V
95		eltg 21854	. . . . . 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 314	. . . 4 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹))))
98	97	pm5.32da 582	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)))))
99		tgtopon 21868	. . . . . 6 ⊢ (𝐽 ∈ TopBases → (topGen‘𝐽) ∈ (TopOn‘∪ 𝐽))
100	99	adantr 484	. . . . 5 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘𝐽) ∈ (TopOn‘∪ 𝐽))
101	18	fveq2i 6720	. . . . 5 ⊢ (TopOn‘𝑋) = (TopOn‘∪ 𝐽)
102	100, 101	eleqtrrdi 2849	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘𝐽) ∈ (TopOn‘𝑋))
103	7	adantl 485	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐹:𝑋–onto→𝑌)
104		elqtop3 22600	. . . 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 587	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ ((topGen‘𝐽) qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽))))
106		unitg 21864	. . . . . . . . 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 22598	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
109	7, 108	sylan2 596	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
110		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) → 𝑥 ⊆ 𝑌)
111		velpw 4518	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝑌 ↔ 𝑥 ⊆ 𝑌)
112	110, 111	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) → 𝑥 ∈ 𝒫 𝑌)
113	109, 112	syl6bi 256	. . . . . . . . . 10 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) → 𝑥 ∈ 𝒫 𝑌))
114	113	ssrdv 3907	. . . . . . . . 9 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝐽 qTop 𝐹) ⊆ 𝒫 𝑌)
115		sspwuni 5008	. . . . . . . . 9 ⊢ ((𝐽 qTop 𝐹) ⊆ 𝒫 𝑌 ↔ ∪ (𝐽 qTop 𝐹) ⊆ 𝑌)
116	114, 115	sylib 221	. . . . . . . 8 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∪ (𝐽 qTop 𝐹) ⊆ 𝑌)
117	107, 116	eqsstrid 3949	. . . . . . 7 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∪ (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝑌)
118		sspwuni 5008	. . . . . . 7 ⊢ ((topGen‘(𝐽 qTop 𝐹)) ⊆ 𝒫 𝑌 ↔ ∪ (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝑌)
119	117, 118	sylibr 237	. . . . . 6 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝒫 𝑌)
120	119	sseld 3900	. . . . 5 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) → 𝑥 ∈ 𝒫 𝑌))
121	120, 111	syl6ib 254	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) → 𝑥 ⊆ 𝑌))
122	121	pm4.71rd 566	. . 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 314	. 2 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ ((topGen‘𝐽) qTop 𝐹) ↔ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹))))
124	123	eqrdv 2735	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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 Vcvv 3408 ∩ cin 3865 ⊆ wss 3866 𝒫 cpw 4513 ∪ cuni 4819 ^◡ccnv 5550 dom cdm 5551 ran crn 5552 “ cima 5554 Fun wfun 6374 Fn wfn 6375 –1-1→wf1 6377 –onto→wfo 6378 –1-1-onto→wf1o 6379 ‘cfv 6380 (class class class)co 7213 topGenctg 16942 qTop cqtop 17008 TopOnctopon 21807 TopBasesctb 21842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-topgen 16948 df-qtop 17012 df-top 21791 df-topon 21808 df-bases 21843

This theorem is referenced by: imasf1oxms 23387

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