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Theorem tgqtop 23741

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 6804	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
2		f1ofun 6793	. . . . . . . . 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 735	. . . . . . 7 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → Fun ^◡𝐹)
5		simpr 487	. . . . . . . 8 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝑥 ⊆ 𝑌)
6		df-rn 5647	. . . . . . . . 9 ⊢ ran 𝐹 = dom ^◡𝐹
7		f1ofo 6799	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
8	7	ad2antlr 735	. . . . . . . . . 10 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝐹:𝑋–onto→𝑌)
9		forn 6766	. . . . . . . . . 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 2801	. . . . . . . 8 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → dom ^◡𝐹 = 𝑌)
12	5, 11	sseqtrrd 3964	. . . . . . 7 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝑥 ⊆ dom ^◡𝐹)
13		funimass4 6916	. . . . . . 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 592	. . . . . 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 3916	. . . . . . 7 ⊢ (𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥))
16		simprl 778	. . . . . . . . . . . . . . . 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 4147	. . . . . . . . . . . . . . 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 23730	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑧 ∈ (𝐽 qTop 𝐹) ↔ (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽)))
20	7, 19	sylan2 601	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑧 ∈ (𝐽 qTop 𝐹) ↔ (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽)))
21	20	ad3antrrr 738	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 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 498	. . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . . . 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 4554	. . . . . . . . . . . . . . 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 6077	. . . . . . . . . . . . . . 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 4551	. . . . . . . . . . . . 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 4143	. . . . . . . . . . . 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 791	. . . . . . . . . . . . . . 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 6792	. . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 7202	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 Fn 𝑌 ∧ 𝑧 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑧) → (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧))
38	33, 35, 36, 37	syl3anc 1382	. . . . . . . . . . . 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 2841	. . . . . . . . . . . . 13 ⊢ (𝑤 = (^◡𝐹 “ 𝑧) → ((^◡𝐹‘𝑦) ∈ 𝑤 ↔ (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧)))
40	39	rspcev 3572	. . . . . . . . . . . 12 ⊢ (((^◡𝐹 “ 𝑧) ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧)) → ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤)
41	29, 38, 40	syl2anc 592	. . . . . . . . . . 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 3154	. . . . . . . . . 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 791	. . . . . . . . . . . . . . . . 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 6793	. . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . . . . 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 4554	. . . . . . . . . . . . . . . 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 6589	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ (^◡𝐹 “ 𝑥)) → (𝐹 “ 𝑤) ⊆ 𝑥)
50	45, 48, 49	syl2anc 592	. . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . 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 6790	. . . . . . . . . . . . . . . . 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 4147	. . . . . . . . . . . . . . . . 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 4887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
57	56, 18	sseqtrrdi 3968	. . . . . . . . . . . . . . . . 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 6808	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑤 ⊆ 𝑋) → (^◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤)
60	54, 58, 59	syl2anc 592	. . . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . 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 23730	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
63	7, 62	sylan2 601	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
64	63	ad3antrrr 738	. . . . . . . . . . . . . 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 721	. . . . . . . . . . . . 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 3448	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
67	66	elpw2 5280	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑤) ∈ 𝒫 𝑥 ↔ (𝐹 “ 𝑤) ⊆ 𝑥)
68	50, 67	sylibr 236	. . . . . . . . . . . . 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 4143	. . . . . . . . . . . 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 3927	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
71	70	adantr 483	. . . . . . . . . . . . . 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 7246	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑦)) = 𝑦)
73	43, 71, 72	syl2anc 592	. . . . . . . . . . . . 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 6792	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 Fn 𝑋)
75	74	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐹 Fn 𝑋)
76	75	ad3antrrr 738	. . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . 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 7202	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ⊆ 𝑋 ∧ (^◡𝐹‘𝑦) ∈ 𝑤) → (𝐹‘(^◡𝐹‘𝑦)) ∈ (𝐹 “ 𝑤))
79	76, 58, 77, 78	syl3anc 1382	. . . . . . . . . . . . 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 2853	. . . . . . . . . . . 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 2841	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹 “ 𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝐹 “ 𝑤)))
82	81	rspcev 3572	. . . . . . . . . . . 12 ⊢ (((𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ (𝐹 “ 𝑤)) → ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧)
83	69, 80, 82	syl2anc 592	. . . . . . . . . . 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 3154	. . . . . . . . . 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 214	. . . . . . . . 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 4859	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧)
87		eluni2 4859	. . . . . . . . 9 ⊢ ((^◡𝐹‘𝑦) ∈ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ↔ ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤)
88	85, 86, 87	3bitr4g 316	. . . . . . . 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 3173	. . . . . . 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 285	. . . . . 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 284	. . . . 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 22986	. . . . . 6 ⊢ (𝐽 ∈ TopBases → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ (^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
93	92	ad2antrr 734	. . . . 5 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ (^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
94		ovex 7414	. . . . . 6 ⊢ (𝐽 qTop 𝐹) ∈ V
95		eltg 22986	. . . . . 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 313	. . . 4 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹))))
98	97	pm5.32da 586	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)))))
99		tgtopon 23000	. . . . . 6 ⊢ (𝐽 ∈ TopBases → (topGen‘𝐽) ∈ (TopOn‘∪ 𝐽))
100	99	adantr 483	. . . . 5 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘𝐽) ∈ (TopOn‘∪ 𝐽))
101	18	fveq2i 6855	. . . . 5 ⊢ (TopOn‘𝑋) = (TopOn‘∪ 𝐽)
102	100, 101	eleqtrrdi 2863	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘𝐽) ∈ (TopOn‘𝑋))
103	7	adantl 484	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐹:𝑋–onto→𝑌)
104		elqtop3 23732	. . . 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 592	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ ((topGen‘𝐽) qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽))))
106		unitg 22996	. . . . . . . . 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 23730	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
109	7, 108	sylan2 601	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
110		velpw 4550	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝑌 ↔ 𝑥 ⊆ 𝑌)
111	110	biranri 508	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) → 𝑥 ∈ 𝒫 𝑌)
112	109, 111	biimtrdi 255	. . . . . . . . . 10 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) → 𝑥 ∈ 𝒫 𝑌))
113	112	ssrdv 3933	. . . . . . . . 9 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝐽 qTop 𝐹) ⊆ 𝒫 𝑌)
114		sspwuni 5047	. . . . . . . . 9 ⊢ ((𝐽 qTop 𝐹) ⊆ 𝒫 𝑌 ↔ ∪ (𝐽 qTop 𝐹) ⊆ 𝑌)
115	113, 114	sylib 220	. . . . . . . 8 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∪ (𝐽 qTop 𝐹) ⊆ 𝑌)
116	107, 115	eqsstrid 3965	. . . . . . 7 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∪ (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝑌)
117		sspwuni 5047	. . . . . . 7 ⊢ ((topGen‘(𝐽 qTop 𝐹)) ⊆ 𝒫 𝑌 ↔ ∪ (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝑌)
118	116, 117	sylibr 236	. . . . . 6 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝒫 𝑌)
119	118	sseld 3926	. . . . 5 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) → 𝑥 ∈ 𝒫 𝑌))
120	119, 110	imbitrdi 253	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) → 𝑥 ⊆ 𝑌))
121	120	pm4.71rd 569	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)))))
122	98, 105, 121	3bitr4d 313	. 2 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ ((topGen‘𝐽) qTop 𝐹) ↔ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹))))
123	122	eqrdv 2750	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 208 ∧ wa 398 = wceq 1550 ∈ wcel 2132 ∀wral 3066 ∃wrex 3076 Vcvv 3444 ∩ cin 3894 ⊆ wss 3895 𝒫 cpw 4545 ∪ cuni 4855 ^◡ccnv 5635 dom cdm 5636 ran crn 5637 “ cima 5639 Fun wfun 6500 Fn wfn 6501 –1-1→wf1 6503 –onto→wfo 6504 –1-1-onto→wf1o 6505 ‘cfv 6506 (class class class)co 7381 topGenctg 17438 qTop cqtop 17505 TopOnctopon 22939 TopBasesctb 22974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-ral 3067 df-rex 3077 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-ov 7384 df-oprab 7385 df-mpo 7386 df-topgen 17444 df-qtop 17509 df-top 22923 df-topon 22940 df-bases 22975

This theorem is referenced by: imasf1oxms 24518

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