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Theorem txcnmpt 23632

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 2737	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
3	1, 2	cnf 23254	. . . . . 6 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹:𝑊⟶∪ 𝑅)
4	3	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹:𝑊⟶∪ 𝑅)
5	4	ffvelcdmda 7104	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐹‘𝑥) ∈ ∪ 𝑅)
6		eqid 2737	. . . . . . 7 ⊢ ∪ 𝑆 = ∪ 𝑆
7	1, 6	cnf 23254	. . . . . 6 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺:𝑊⟶∪ 𝑆)
8	7	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺:𝑊⟶∪ 𝑆)
9	8	ffvelcdmda 7104	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐺‘𝑥) ∈ ∪ 𝑆)
10	5, 9	opelxpd 5724	. . 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 7134	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻:𝑊⟶(∪ 𝑅 × ∪ 𝑆))
13	11	mptpreima 6258	. . . . . 6 ⊢ (^◡𝐻 “ (𝑟 × 𝑠)) = {𝑥 ∈ 𝑊 ∣ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠)}
14	4	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐹:𝑊⟶∪ 𝑅)
15	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐹:𝑊⟶∪ 𝑅)
16		ffn 6736	. . . . . . . . . . . 12 ⊢ (𝐹:𝑊⟶∪ 𝑅 → 𝐹 Fn 𝑊)
17		elpreima 7078	. . . . . . . . . . . 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 528	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → ((𝐹‘𝑥) ∈ 𝑟 ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟)))
20	19	adantl 481	. . . . . . . . . . 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 282	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (^◡𝐹 “ 𝑟) ↔ (𝐹‘𝑥) ∈ 𝑟))
22	8	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐺:𝑊⟶∪ 𝑆)
23		ffn 6736	. . . . . . . . . . . 12 ⊢ (𝐺:𝑊⟶∪ 𝑆 → 𝐺 Fn 𝑊)
24		elpreima 7078	. . . . . . . . . . . 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 528	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → ((𝐺‘𝑥) ∈ 𝑠 ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠)))
27	26	adantl 481	. . . . . . . . . . 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 282	. . . . . . . . . 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 3967	. . . . . . . . 9 ⊢ (𝑥 ∈ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑟) ∧ 𝑥 ∈ (^◡𝐺 “ 𝑠)))
31		opelxp 5721	. . . . . . . . 9 ⊢ (⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠) ↔ ((𝐹‘𝑥) ∈ 𝑟 ∧ (𝐺‘𝑥) ∈ 𝑠))
32	29, 30, 31	3bitr4g 314	. . . . . . . 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 4226	. . . . . . 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 4237	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ (^◡𝐹 “ 𝑟)
35		cnvimass 6100	. . . . . . . . . 10 ⊢ (^◡𝐹 “ 𝑟) ⊆ dom 𝐹
36	34, 35	sstri 3993	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ dom 𝐹
37	36, 14	fssdm 6755	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ 𝑊)
38		sseqin2 4223	. . . . . . . 8 ⊢ (((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ 𝑊 ↔ (𝑊 ∩ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠))) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
39	37, 38	sylib 218	. . . . . . 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 2779	. . . . . 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	eqtrid 2789	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
42		cntop1 23248	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑈 ∈ Top)
43	42	adantl 481	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ Top)
44	43	adantr 480	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑈 ∈ Top)
45		cnima 23273	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝑟 ∈ 𝑅) → (^◡𝐹 “ 𝑟) ∈ 𝑈)
46	45	ad2ant2r 747	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐹 “ 𝑟) ∈ 𝑈)
47		cnima 23273	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑈 Cn 𝑆) ∧ 𝑠 ∈ 𝑆) → (^◡𝐺 “ 𝑠) ∈ 𝑈)
48	47	ad2ant2l 746	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐺 “ 𝑠) ∈ 𝑈)
49		inopn 22905	. . . . . 6 ⊢ ((𝑈 ∈ Top ∧ (^◡𝐹 “ 𝑟) ∈ 𝑈 ∧ (^◡𝐺 “ 𝑠) ∈ 𝑈) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
50	44, 46, 48, 49	syl3anc 1373	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
51	41, 50	eqeltrd 2841	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
52	51	ralrimivva 3202	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
53		vex 3484	. . . . . 6 ⊢ 𝑟 ∈ V
54		vex 3484	. . . . . 6 ⊢ 𝑠 ∈ V
55	53, 54	xpex 7773	. . . . 5 ⊢ (𝑟 × 𝑠) ∈ V
56	55	rgen2w 3066	. . . 4 ⊢ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V
57		eqid 2737	. . . . 5 ⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
58		imaeq2 6074	. . . . . 6 ⊢ (𝑧 = (𝑟 × 𝑠) → (^◡𝐻 “ 𝑧) = (^◡𝐻 “ (𝑟 × 𝑠)))
59	58	eleq1d 2826	. . . . 5 ⊢ (𝑧 = (𝑟 × 𝑠) → ((^◡𝐻 “ 𝑧) ∈ 𝑈 ↔ (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈))
60	57, 59	ralrnmpo 7572	. . . 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 234	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑧 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(^◡𝐻 “ 𝑧) ∈ 𝑈)
63	1	toptopon 22923	. . . 4 ⊢ (𝑈 ∈ Top ↔ 𝑈 ∈ (TopOn‘𝑊))
64	43, 63	sylib 218	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ (TopOn‘𝑊))
65		cntop2 23249	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
66		cntop2 23249	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
67		eqid 2737	. . . . 5 ⊢ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
68	67	txval 23572	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
69	65, 66, 68	syl2an 596	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×_t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
70		toptopon2 22924	. . . . 5 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
71	65, 70	sylib 218	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ (TopOn‘∪ 𝑅))
72		toptopon2 22924	. . . . 5 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
73	66, 72	sylib 218	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ (TopOn‘∪ 𝑆))
74		txtopon 23599	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
75	71, 73, 74	syl2an 596	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
76	64, 69, 75	tgcn 23260	. 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 713	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ⟨cop 4632 ∪ cuni 4907 ↦ cmpt 5225 × cxp 5683 ^◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 topGenctg 17482 Topctop 22899 TopOnctopon 22916 Cn ccn 23232 ×_t ctx 23568

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cn 23235 df-tx 23570

This theorem is referenced by: uptx 23633 hauseqlcld 23654 txkgen 23660 cnmpt1t 23673 cnmpt2t 23681 txpconn 35237

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