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Theorem txcnmpt 14987

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 2229	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
3	1, 2	cnf 14918	. . . . . 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 5778	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐹‘𝑥) ∈ ∪ 𝑅)
6		eqid 2229	. . . . . . 7 ⊢ ∪ 𝑆 = ∪ 𝑆
7	1, 6	cnf 14918	. . . . . 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 5778	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐺‘𝑥) ∈ ∪ 𝑆)
10	5, 9	opelxpd 4756	. . 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 5797	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻:𝑊⟶(∪ 𝑅 × ∪ 𝑆))
13	11	mptpreima 5228	. . . . . 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 5479	. . . . . . . . . . . 12 ⊢ (𝐹:𝑊⟶∪ 𝑅 → 𝐹 Fn 𝑊)
17		elpreima 5762	. . . . . . . . . . . 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 5479	. . . . . . . . . . . 12 ⊢ (𝐺:𝑊⟶∪ 𝑆 → 𝐺 Fn 𝑊)
24		elpreima 5762	. . . . . . . . . . . 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 3388	. . . . . . . . 9 ⊢ (𝑥 ∈ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑟) ∧ 𝑥 ∈ (^◡𝐺 “ 𝑠)))
31		opelxp 4753	. . . . . . . . 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 3413	. . . . . . 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 3425	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ (^◡𝐹 “ 𝑟)
35		cnvimass 5097	. . . . . . . . . 10 ⊢ (^◡𝐹 “ 𝑟) ⊆ dom 𝐹
36	34, 35	sstri 3234	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ dom 𝐹
37	36, 14	fssdm 5494	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ 𝑊)
38		sseqin2 3424	. . . . . . . 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 2264	. . . . . 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 2274	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
42		cntop1 14915	. . . . . . . 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 14934	. . . . . . 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 14934	. . . . . . 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 14717	. . . . . 6 ⊢ ((𝑈 ∈ Top ∧ (^◡𝐹 “ 𝑟) ∈ 𝑈 ∧ (^◡𝐺 “ 𝑠) ∈ 𝑈) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
50	44, 46, 48, 49	syl3anc 1271	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
51	41, 50	eqeltrd 2306	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
52	51	ralrimivva 2612	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
53		vex 2803	. . . . . 6 ⊢ 𝑟 ∈ V
54		vex 2803	. . . . . 6 ⊢ 𝑠 ∈ V
55	53, 54	xpex 4840	. . . . 5 ⊢ (𝑟 × 𝑠) ∈ V
56	55	rgen2w 2586	. . . 4 ⊢ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V
57		eqid 2229	. . . . 5 ⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
58		imaeq2 5070	. . . . . 6 ⊢ (𝑧 = (𝑟 × 𝑠) → (^◡𝐻 “ 𝑧) = (^◡𝐻 “ (𝑟 × 𝑠)))
59	58	eleq1d 2298	. . . . 5 ⊢ (𝑧 = (𝑟 × 𝑠) → ((^◡𝐻 “ 𝑧) ∈ 𝑈 ↔ (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈))
60	57, 59	ralrnmpo 6131	. . . 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 14732	. . . 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 14916	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
66		cntop2 14916	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
67		eqid 2229	. . . . 5 ⊢ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
68	67	txval 14969	. . . 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 14733	. . . . 5 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
71	65, 70	sylib 122	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ (TopOn‘∪ 𝑅))
72		toptopon2 14733	. . . . 5 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
73	66, 72	sylib 122	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ (TopOn‘∪ 𝑆))
74		txtopon 14976	. . . 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 14922	. 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 950	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 1395 ∈ wcel 2200 ∀wral 2508 {crab 2512 Vcvv 2800 ∩ cin 3197 ⊆ wss 3198 ⟨cop 3670 ∪ cuni 3891 ↦ cmpt 4148 × cxp 4721 ^◡ccnv 4722 dom cdm 4723 ran crn 4724 “ cima 4726 Fn wfn 5319 ⟶wf 5320 ‘cfv 5324 (class class class)co 6013 ∈ cmpo 6015 topGenctg 13327 Topctop 14711 TopOnctopon 14724 Cn ccn 14899 ×_t ctx 14966

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-map 6814 df-topgen 13333 df-top 14712 df-topon 14725 df-bases 14757 df-cn 14902 df-tx 14967

This theorem is referenced by: uptx 14988 cnmpt1t 14999 cnmpt2t 15007

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