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Theorem txcnmpt 15084

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 2231	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
3	1, 2	cnf 15015	. . . . . 6 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹:𝑊⟶∪ 𝑅)
4	3	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹:𝑊⟶∪ 𝑅)
5	4	ffvelcdmda 5790	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐹‘𝑥) ∈ ∪ 𝑅)
6		eqid 2231	. . . . . . 7 ⊢ ∪ 𝑆 = ∪ 𝑆
7	1, 6	cnf 15015	. . . . . 6 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺:𝑊⟶∪ 𝑆)
8	7	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺:𝑊⟶∪ 𝑆)
9	8	ffvelcdmda 5790	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐺‘𝑥) ∈ ∪ 𝑆)
10	5, 9	opelxpd 4764	. . 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 5809	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻:𝑊⟶(∪ 𝑅 × ∪ 𝑆))
13	11	mptpreima 5237	. . . . . 6 ⊢ (^◡𝐻 “ (𝑟 × 𝑠)) = {𝑥 ∈ 𝑊 ∣ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠)}
14	4	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐹:𝑊⟶∪ 𝑅)
15	14	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐹:𝑊⟶∪ 𝑅)
16		ffn 5489	. . . . . . . . . . . 12 ⊢ (𝐹:𝑊⟶∪ 𝑅 → 𝐹 Fn 𝑊)
17		elpreima 5775	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑊 → (𝑥 ∈ (^◡𝐹 “ 𝑟) ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟)))
18	15, 16, 17	3syl 17	. . . . . . . . . . 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 301	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → ((𝐹‘𝑥) ∈ 𝑟 ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟)))
20	19	adantl 277	. . . . . . . . . . 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 191	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (^◡𝐹 “ 𝑟) ↔ (𝐹‘𝑥) ∈ 𝑟))
22	8	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐺:𝑊⟶∪ 𝑆)
23		ffn 5489	. . . . . . . . . . . 12 ⊢ (𝐺:𝑊⟶∪ 𝑆 → 𝐺 Fn 𝑊)
24		elpreima 5775	. . . . . . . . . . . 12 ⊢ (𝐺 Fn 𝑊 → (𝑥 ∈ (^◡𝐺 “ 𝑠) ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠)))
25	22, 23, 24	3syl 17	. . . . . . . . . . 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 301	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → ((𝐺‘𝑥) ∈ 𝑠 ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠)))
27	26	adantl 277	. . . . . . . . . . 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 191	. . . . . . . . . 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 473	. . . . . . . . 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 3392	. . . . . . . . 9 ⊢ (𝑥 ∈ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑟) ∧ 𝑥 ∈ (^◡𝐺 “ 𝑠)))
31		opelxp 4761	. . . . . . . . 9 ⊢ (⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠) ↔ ((𝐹‘𝑥) ∈ 𝑟 ∧ (𝐺‘𝑥) ∈ 𝑠))
32	29, 30, 31	3bitr4g 223	. . . . . . . 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 3417	. . . . . . 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 3429	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ (^◡𝐹 “ 𝑟)
35		cnvimass 5106	. . . . . . . . . 10 ⊢ (^◡𝐹 “ 𝑟) ⊆ dom 𝐹
36	34, 35	sstri 3237	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ dom 𝐹
37	36, 14	fssdm 5504	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ 𝑊)
38		sseqin2 3428	. . . . . . . 8 ⊢ (((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ 𝑊 ↔ (𝑊 ∩ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠))) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
39	37, 38	sylib 122	. . . . . . 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 2266	. . . . . 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 2276	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
42		cntop1 15012	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑈 ∈ Top)
43	42	adantl 277	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ Top)
44	43	adantr 276	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑈 ∈ Top)
45		cnima 15031	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝑟 ∈ 𝑅) → (^◡𝐹 “ 𝑟) ∈ 𝑈)
46	45	ad2ant2r 509	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐹 “ 𝑟) ∈ 𝑈)
47		cnima 15031	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑈 Cn 𝑆) ∧ 𝑠 ∈ 𝑆) → (^◡𝐺 “ 𝑠) ∈ 𝑈)
48	47	ad2ant2l 508	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐺 “ 𝑠) ∈ 𝑈)
49		inopn 14814	. . . . . 6 ⊢ ((𝑈 ∈ Top ∧ (^◡𝐹 “ 𝑟) ∈ 𝑈 ∧ (^◡𝐺 “ 𝑠) ∈ 𝑈) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
50	44, 46, 48, 49	syl3anc 1274	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
51	41, 50	eqeltrd 2308	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
52	51	ralrimivva 2615	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
53		vex 2806	. . . . . 6 ⊢ 𝑟 ∈ V
54		vex 2806	. . . . . 6 ⊢ 𝑠 ∈ V
55	53, 54	xpex 4848	. . . . 5 ⊢ (𝑟 × 𝑠) ∈ V
56	55	rgen2w 2589	. . . 4 ⊢ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V
57		eqid 2231	. . . . 5 ⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
58		imaeq2 5078	. . . . . 6 ⊢ (𝑧 = (𝑟 × 𝑠) → (^◡𝐻 “ 𝑧) = (^◡𝐻 “ (𝑟 × 𝑠)))
59	58	eleq1d 2300	. . . . 5 ⊢ (𝑧 = (𝑟 × 𝑠) → ((^◡𝐻 “ 𝑧) ∈ 𝑈 ↔ (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈))
60	57, 59	ralrnmpo 6146	. . . 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 134	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑧 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(^◡𝐻 “ 𝑧) ∈ 𝑈)
63	1	toptopon 14829	. . . 4 ⊢ (𝑈 ∈ Top ↔ 𝑈 ∈ (TopOn‘𝑊))
64	43, 63	sylib 122	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ (TopOn‘𝑊))
65		cntop2 15013	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
66		cntop2 15013	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
67		eqid 2231	. . . . 5 ⊢ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
68	67	txval 15066	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
69	65, 66, 68	syl2an 289	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×_t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
70		toptopon2 14830	. . . . 5 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
71	65, 70	sylib 122	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ (TopOn‘∪ 𝑅))
72		toptopon2 14830	. . . . 5 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
73	66, 72	sylib 122	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ (TopOn‘∪ 𝑆))
74		txtopon 15073	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
75	71, 73, 74	syl2an 289	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
76	64, 69, 75	tgcn 15019	. 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 953	1 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻 ∈ (𝑈 Cn (𝑅 ×_t 𝑆)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2202 ∀wral 2511 {crab 2515 Vcvv 2803 ∩ cin 3200 ⊆ wss 3201 ⟨cop 3676 ∪ cuni 3898 ↦ cmpt 4155 × cxp 4729 ^◡ccnv 4730 dom cdm 4731 ran crn 4732 “ cima 4734 Fn wfn 5328 ⟶wf 5329 ‘cfv 5333 (class class class)co 6028 ∈ cmpo 6030 topGenctg 13417 Topctop 14808 TopOnctopon 14821 Cn ccn 14996 ×_t ctx 15063

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-map 6862 df-topgen 13423 df-top 14809 df-topon 14822 df-bases 14854 df-cn 14999 df-tx 15064

This theorem is referenced by: uptx 15085 cnmpt1t 15096 cnmpt2t 15104

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