tgqtop - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tgqtop		Structured version Visualization version GIF version

Theorem tgqtop 22771

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 6712	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
2		f1ofun 6702	. . . . . . . . 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 723	. . . . . . 7 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → Fun ^◡𝐹)
5		simpr 484	. . . . . . . 8 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝑥 ⊆ 𝑌)
6		df-rn 5591	. . . . . . . . 9 ⊢ ran 𝐹 = dom ^◡𝐹
7		f1ofo 6707	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
8	7	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝐹:𝑋–onto→𝑌)
9		forn 6675	. . . . . . . . . 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 2793	. . . . . . . 8 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → dom ^◡𝐹 = 𝑌)
12	5, 11	sseqtrrd 3958	. . . . . . 7 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → 𝑥 ⊆ dom ^◡𝐹)
13		funimass4 6816	. . . . . . 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 583	. . . . . 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 3905	. . . . . . 7 ⊢ (𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥))
16		simprl 767	. . . . . . . . . . . . . . . 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 4128	. . . . . . . . . . . . . . 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 22760	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑧 ∈ (𝐽 qTop 𝐹) ↔ (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽)))
20	7, 19	sylan2 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑧 ∈ (𝐽 qTop 𝐹) ↔ (𝑧 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑧) ∈ 𝐽)))
21	20	ad3antrrr 726	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . 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 4541	. . . . . . . . . . . . . . 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 5999	. . . . . . . . . . . . . . 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 4538	. . . . . . . . . . . . 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 4124	. . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 6701	. . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . 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 3927	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 7091	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 Fn 𝑌 ∧ 𝑧 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑧) → (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧))
38	33, 35, 36, 37	syl3anc 1369	. . . . . . . . . . . 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 3552	. . . . . . . . . . . 12 ⊢ (((^◡𝐹 “ 𝑧) ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ∧ (^◡𝐹‘𝑦) ∈ (^◡𝐹 “ 𝑧)) → ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤)
41	29, 38, 40	syl2anc 583	. . . . . . . . . . 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 3213	. . . . . . . . . 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 780	. . . . . . . . . . . . . . . . 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 6702	. . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . . 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 4541	. . . . . . . . . . . . . . . 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 6501	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ (^◡𝐹 “ 𝑥)) → (𝐹 “ 𝑤) ⊆ 𝑥)
50	45, 48, 49	syl2anc 583	. . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . 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 3927	. . . . . . . . . . . . . 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 6699	. . . . . . . . . . . . . . . . 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 4128	. . . . . . . . . . . . . . . . 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 4868	. . . . . . . . . . . . . . . . . 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 6716	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑤 ⊆ 𝑋) → (^◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤)
60	54, 58, 59	syl2anc 583	. . . . . . . . . . . . . . 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 22760	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
63	7, 62	sylan2 592	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (^◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽)))
64	63	ad3antrrr 726	. . . . . . . . . . . . . 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 709	. . . . . . . . . . . . 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 3426	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
67	66	elpw2 5264	. . . . . . . . . . . . . 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 4124	. . . . . . . . . . . 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 3917	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
71	70	adantr 480	. . . . . . . . . . . . . 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 7130	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑦)) = 𝑦)
73	43, 71, 72	syl2anc 583	. . . . . . . . . . . . 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 6701	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 Fn 𝑋)
75	74	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐹 Fn 𝑋)
76	75	ad3antrrr 726	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . 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 7091	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ⊆ 𝑋 ∧ (^◡𝐹‘𝑦) ∈ 𝑤) → (𝐹‘(^◡𝐹‘𝑦)) ∈ (𝐹 “ 𝑤))
79	76, 58, 77, 78	syl3anc 1369	. . . . . . . . . . . . 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 3552	. . . . . . . . . . . 12 ⊢ (((𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ∧ 𝑦 ∈ (𝐹 “ 𝑤)) → ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧)
83	69, 80, 82	syl2anc 583	. . . . . . . . . . 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 3213	. . . . . . . . . 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 4840	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥) ↔ ∃𝑧 ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 𝑥)𝑦 ∈ 𝑧)
87		eluni2 4840	. . . . . . . . 9 ⊢ ((^◡𝐹‘𝑦) ∈ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥)) ↔ ∃𝑤 ∈ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))(^◡𝐹‘𝑦) ∈ 𝑤)
88	85, 86, 87	3bitr4g 313	. . . . . . . 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 3119	. . . . . . 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 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 22015	. . . . . 6 ⊢ (𝐽 ∈ TopBases → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ (^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
93	92	ad2antrr 722	. . . . 5 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ (^◡𝐹 “ 𝑥) ⊆ ∪ (𝐽 ∩ 𝒫 (^◡𝐹 “ 𝑥))))
94		ovex 7288	. . . . . 6 ⊢ (𝐽 qTop 𝐹) ∈ V
95		eltg 22015	. . . . . 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 310	. . . 4 ⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑥 ⊆ 𝑌) → ((^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽) ↔ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹))))
98	97	pm5.32da 578	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 ∈ (topGen‘(𝐽 qTop 𝐹)))))
99		tgtopon 22029	. . . . . 6 ⊢ (𝐽 ∈ TopBases → (topGen‘𝐽) ∈ (TopOn‘∪ 𝐽))
100	99	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (topGen‘𝐽) ∈ (TopOn‘∪ 𝐽))
101	18	fveq2i 6759	. . . . 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 481	. . . 4 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐹:𝑋–onto→𝑌)
104		elqtop3 22762	. . . 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 583	. . 3 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ ((topGen‘𝐽) qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ (topGen‘𝐽))))
106		unitg 22025	. . . . . . . . 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 22760	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
109	7, 108	sylan2 592	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽)))
110		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝑌 ∧ (^◡𝐹 “ 𝑥) ∈ 𝐽) → 𝑥 ⊆ 𝑌)
111		velpw 4535	. . . . . . . . . . . 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 3923	. . . . . . . . 9 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝐽 qTop 𝐹) ⊆ 𝒫 𝑌)
115		sspwuni 5025	. . . . . . . . 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 3965	. . . . . . 7 ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∪ (topGen‘(𝐽 qTop 𝐹)) ⊆ 𝑌)
118		sspwuni 5025	. . . . . . 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 3916	. . . . 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 562	. . 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 310	. 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 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 𝒫 cpw 4530 ∪ cuni 4836 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 “ cima 5583 Fun wfun 6412 Fn wfn 6413 –1-1→wf1 6415 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 topGenctg 17065 qTop cqtop 17131 TopOnctopon 21967 TopBasesctb 22003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-topgen 17071 df-qtop 17135 df-top 21951 df-topon 21968 df-bases 22004

This theorem is referenced by: imasf1oxms 23551

W3C validator