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Theorem txcnmpt 22224

Description: A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)

**Hypotheses**
Ref	Expression
txcnmpt.1	⊢ 𝑊 = ∪ 𝑈
txcnmpt.2	⊢ 𝐻 = (𝑥 ∈ 𝑊 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)

**Assertion**
Ref	Expression
txcnmpt	⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻 ∈ (𝑈 Cn (𝑅 ×_t 𝑆)))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐺 𝑥,𝑅 𝑥,𝑆 𝑥,𝑈 𝑥,𝑊 Allowed substitution hint: 𝐻(𝑥)

Proof of Theorem txcnmpt Dummy variables 𝑠 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txcnmpt.1	. . . . . . 7 ⊢ 𝑊 = ∪ 𝑈
2		eqid 2819	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
3	1, 2	cnf 21846	. . . . . 6 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹:𝑊⟶∪ 𝑅)
4	3	adantr 483	. . . . 5 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹:𝑊⟶∪ 𝑅)
5	4	ffvelrnda 6844	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐹‘𝑥) ∈ ∪ 𝑅)
6		eqid 2819	. . . . . . 7 ⊢ ∪ 𝑆 = ∪ 𝑆
7	1, 6	cnf 21846	. . . . . 6 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺:𝑊⟶∪ 𝑆)
8	7	adantl 484	. . . . 5 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺:𝑊⟶∪ 𝑆)
9	8	ffvelrnda 6844	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐺‘𝑥) ∈ ∪ 𝑆)
10	5, 9	opelxpd 5586	. . 3 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (∪ 𝑅 × ∪ 𝑆))
11		txcnmpt.2	. . 3 ⊢ 𝐻 = (𝑥 ∈ 𝑊 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
12	10, 11	fmptd 6871	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻:𝑊⟶(∪ 𝑅 × ∪ 𝑆))
13	11	mptpreima 6085	. . . . . 6 ⊢ (^◡𝐻 “ (𝑟 × 𝑠)) = {𝑥 ∈ 𝑊 ∣ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠)}
14	4	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐹:𝑊⟶∪ 𝑅)
15	14	adantr 483	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐹:𝑊⟶∪ 𝑅)
16		ffn 6507	. . . . . . . . . . . 12 ⊢ (𝐹:𝑊⟶∪ 𝑅 → 𝐹 Fn 𝑊)
17		elpreima 6821	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑊 → (𝑥 ∈ (^◡𝐹 “ 𝑟) ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟)))
18	15, 16, 17	3syl 18	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (^◡𝐹 “ 𝑟) ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟)))
19		ibar 531	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → ((𝐹‘𝑥) ∈ 𝑟 ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟)))
20	19	adantl 484	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → ((𝐹‘𝑥) ∈ 𝑟 ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟)))
21	18, 20	bitr4d 284	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (^◡𝐹 “ 𝑟) ↔ (𝐹‘𝑥) ∈ 𝑟))
22	8	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐺:𝑊⟶∪ 𝑆)
23		ffn 6507	. . . . . . . . . . . 12 ⊢ (𝐺:𝑊⟶∪ 𝑆 → 𝐺 Fn 𝑊)
24		elpreima 6821	. . . . . . . . . . . 12 ⊢ (𝐺 Fn 𝑊 → (𝑥 ∈ (^◡𝐺 “ 𝑠) ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠)))
25	22, 23, 24	3syl 18	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (^◡𝐺 “ 𝑠) ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠)))
26		ibar 531	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → ((𝐺‘𝑥) ∈ 𝑠 ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠)))
27	26	adantl 484	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → ((𝐺‘𝑥) ∈ 𝑠 ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠)))
28	25, 27	bitr4d 284	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (^◡𝐺 “ 𝑠) ↔ (𝐺‘𝑥) ∈ 𝑠))
29	21, 28	anbi12d 632	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → ((𝑥 ∈ (^◡𝐹 “ 𝑟) ∧ 𝑥 ∈ (^◡𝐺 “ 𝑠)) ↔ ((𝐹‘𝑥) ∈ 𝑟 ∧ (𝐺‘𝑥) ∈ 𝑠)))
30		elin 4167	. . . . . . . . 9 ⊢ (𝑥 ∈ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑟) ∧ 𝑥 ∈ (^◡𝐺 “ 𝑠)))
31		opelxp 5584	. . . . . . . . 9 ⊢ (⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠) ↔ ((𝐹‘𝑥) ∈ 𝑟 ∧ (𝐺‘𝑥) ∈ 𝑠))
32	29, 30, 31	3bitr4g 316	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ↔ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠)))
33	32	rabbi2dva 4192	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑊 ∩ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠))) = {𝑥 ∈ 𝑊 ∣ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠)})
34		inss1 4203	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ (^◡𝐹 “ 𝑟)
35		cnvimass 5942	. . . . . . . . . 10 ⊢ (^◡𝐹 “ 𝑟) ⊆ dom 𝐹
36	34, 35	sstri 3974	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ dom 𝐹
37	36, 14	fssdm 6523	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ 𝑊)
38		sseqin2 4190	. . . . . . . 8 ⊢ (((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ 𝑊 ↔ (𝑊 ∩ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠))) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
39	37, 38	sylib 220	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑊 ∩ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠))) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
40	33, 39	eqtr3d 2856	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → {𝑥 ∈ 𝑊 ∣ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠)} = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
41	13, 40	syl5eq 2866	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
42		cntop1 21840	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑈 ∈ Top)
43	42	adantl 484	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ Top)
44	43	adantr 483	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑈 ∈ Top)
45		cnima 21865	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝑟 ∈ 𝑅) → (^◡𝐹 “ 𝑟) ∈ 𝑈)
46	45	ad2ant2r 745	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐹 “ 𝑟) ∈ 𝑈)
47		cnima 21865	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑈 Cn 𝑆) ∧ 𝑠 ∈ 𝑆) → (^◡𝐺 “ 𝑠) ∈ 𝑈)
48	47	ad2ant2l 744	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐺 “ 𝑠) ∈ 𝑈)
49		inopn 21499	. . . . . 6 ⊢ ((𝑈 ∈ Top ∧ (^◡𝐹 “ 𝑟) ∈ 𝑈 ∧ (^◡𝐺 “ 𝑠) ∈ 𝑈) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
50	44, 46, 48, 49	syl3anc 1366	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
51	41, 50	eqeltrd 2911	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
52	51	ralrimivva 3189	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
53		vex 3496	. . . . . 6 ⊢ 𝑟 ∈ V
54		vex 3496	. . . . . 6 ⊢ 𝑠 ∈ V
55	53, 54	xpex 7468	. . . . 5 ⊢ (𝑟 × 𝑠) ∈ V
56	55	rgen2w 3149	. . . 4 ⊢ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V
57		eqid 2819	. . . . 5 ⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
58		imaeq2 5918	. . . . . 6 ⊢ (𝑧 = (𝑟 × 𝑠) → (^◡𝐻 “ 𝑧) = (^◡𝐻 “ (𝑟 × 𝑠)))
59	58	eleq1d 2895	. . . . 5 ⊢ (𝑧 = (𝑟 × 𝑠) → ((^◡𝐻 “ 𝑧) ∈ 𝑈 ↔ (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈))
60	57, 59	ralrnmpo 7281	. . . 4 ⊢ (∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V → (∀𝑧 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(^◡𝐻 “ 𝑧) ∈ 𝑈 ↔ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈))
61	56, 60	ax-mp 5	. . 3 ⊢ (∀𝑧 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(^◡𝐻 “ 𝑧) ∈ 𝑈 ↔ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
62	52, 61	sylibr 236	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑧 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(^◡𝐻 “ 𝑧) ∈ 𝑈)
63	1	toptopon 21517	. . . 4 ⊢ (𝑈 ∈ Top ↔ 𝑈 ∈ (TopOn‘𝑊))
64	43, 63	sylib 220	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ (TopOn‘𝑊))
65		cntop2 21841	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
66		cntop2 21841	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
67		eqid 2819	. . . . 5 ⊢ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
68	67	txval 22164	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
69	65, 66, 68	syl2an 597	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×_t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
70		toptopon2 21518	. . . . 5 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
71	65, 70	sylib 220	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ (TopOn‘∪ 𝑅))
72		toptopon2 21518	. . . . 5 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
73	66, 72	sylib 220	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ (TopOn‘∪ 𝑆))
74		txtopon 22191	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
75	71, 73, 74	syl2an 597	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
76	64, 69, 75	tgcn 21852	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝐻 ∈ (𝑈 Cn (𝑅 ×_t 𝑆)) ↔ (𝐻:𝑊⟶(∪ 𝑅 × ∪ 𝑆) ∧ ∀𝑧 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(^◡𝐻 “ 𝑧) ∈ 𝑈)))
77	12, 62, 76	mpbir2and 711	1 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻 ∈ (𝑈 Cn (𝑅 ×_t 𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 {crab 3140 Vcvv 3493 ∩ cin 3933 ⊆ wss 3934 ⟨cop 4565 ∪ cuni 4830 ↦ cmpt 5137 × cxp 5546 ^◡ccnv 5547 dom cdm 5548 ran crn 5549 “ cima 5551 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ∈ cmpo 7150 topGenctg 16703 Topctop 21493 TopOnctopon 21510 Cn ccn 21824 ×_t ctx 22160

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-map 8400 df-topgen 16709 df-top 21494 df-topon 21511 df-bases 21546 df-cn 21827 df-tx 22162

This theorem is referenced by: uptx 22225 hauseqlcld 22246 txkgen 22252 cnmpt1t 22265 cnmpt2t 22273 txpconn 32472

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