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Theorem txcnmpt 13067

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 2170	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
3	1, 2	cnf 12998	. . . . . 6 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹:𝑊⟶∪ 𝑅)
4	3	adantr 274	. . . . 5 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹:𝑊⟶∪ 𝑅)
5	4	ffvelrnda 5631	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐹‘𝑥) ∈ ∪ 𝑅)
6		eqid 2170	. . . . . . 7 ⊢ ∪ 𝑆 = ∪ 𝑆
7	1, 6	cnf 12998	. . . . . 6 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺:𝑊⟶∪ 𝑆)
8	7	adantl 275	. . . . 5 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺:𝑊⟶∪ 𝑆)
9	8	ffvelrnda 5631	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐺‘𝑥) ∈ ∪ 𝑆)
10	5, 9	opelxpd 4644	. . 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 5650	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻:𝑊⟶(∪ 𝑅 × ∪ 𝑆))
13	11	mptpreima 5104	. . . . . 6 ⊢ (^◡𝐻 “ (𝑟 × 𝑠)) = {𝑥 ∈ 𝑊 ∣ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠)}
14	4	adantr 274	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐹:𝑊⟶∪ 𝑅)
15	14	adantr 274	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐹:𝑊⟶∪ 𝑅)
16		ffn 5347	. . . . . . . . . . . 12 ⊢ (𝐹:𝑊⟶∪ 𝑅 → 𝐹 Fn 𝑊)
17		elpreima 5615	. . . . . . . . . . . 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 299	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → ((𝐹‘𝑥) ∈ 𝑟 ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟)))
20	19	adantl 275	. . . . . . . . . . 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 190	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (^◡𝐹 “ 𝑟) ↔ (𝐹‘𝑥) ∈ 𝑟))
22	8	ad2antrr 485	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐺:𝑊⟶∪ 𝑆)
23		ffn 5347	. . . . . . . . . . . 12 ⊢ (𝐺:𝑊⟶∪ 𝑆 → 𝐺 Fn 𝑊)
24		elpreima 5615	. . . . . . . . . . . 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 299	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → ((𝐺‘𝑥) ∈ 𝑠 ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠)))
27	26	adantl 275	. . . . . . . . . . 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 190	. . . . . . . . . 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 470	. . . . . . . . 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 3310	. . . . . . . . 9 ⊢ (𝑥 ∈ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑟) ∧ 𝑥 ∈ (^◡𝐺 “ 𝑠)))
31		opelxp 4641	. . . . . . . . 9 ⊢ (⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑟 × 𝑠) ↔ ((𝐹‘𝑥) ∈ 𝑟 ∧ (𝐺‘𝑥) ∈ 𝑠))
32	29, 30, 31	3bitr4g 222	. . . . . . . 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 3335	. . . . . . 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 3347	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ (^◡𝐹 “ 𝑟)
35		cnvimass 4974	. . . . . . . . . 10 ⊢ (^◡𝐹 “ 𝑟) ⊆ dom 𝐹
36	34, 35	sstri 3156	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ dom 𝐹
37	36, 14	fssdm 5362	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ 𝑊)
38		sseqin2 3346	. . . . . . . 8 ⊢ (((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ⊆ 𝑊 ↔ (𝑊 ∩ ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠))) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
39	37, 38	sylib 121	. . . . . . 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 2205	. . . . . 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 2215	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) = ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)))
42		cntop1 12995	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑈 ∈ Top)
43	42	adantl 275	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ Top)
44	43	adantr 274	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑈 ∈ Top)
45		cnima 13014	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝑟 ∈ 𝑅) → (^◡𝐹 “ 𝑟) ∈ 𝑈)
46	45	ad2ant2r 506	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐹 “ 𝑟) ∈ 𝑈)
47		cnima 13014	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑈 Cn 𝑆) ∧ 𝑠 ∈ 𝑆) → (^◡𝐺 “ 𝑠) ∈ 𝑈)
48	47	ad2ant2l 505	. . . . . 6 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐺 “ 𝑠) ∈ 𝑈)
49		inopn 12795	. . . . . 6 ⊢ ((𝑈 ∈ Top ∧ (^◡𝐹 “ 𝑟) ∈ 𝑈 ∧ (^◡𝐺 “ 𝑠) ∈ 𝑈) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
50	44, 46, 48, 49	syl3anc 1233	. . . . 5 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((^◡𝐹 “ 𝑟) ∩ (^◡𝐺 “ 𝑠)) ∈ 𝑈)
51	41, 50	eqeltrd 2247	. . . 4 ⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
52	51	ralrimivva 2552	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)
53		vex 2733	. . . . . 6 ⊢ 𝑟 ∈ V
54		vex 2733	. . . . . 6 ⊢ 𝑠 ∈ V
55	53, 54	xpex 4726	. . . . 5 ⊢ (𝑟 × 𝑠) ∈ V
56	55	rgen2w 2526	. . . 4 ⊢ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V
57		eqid 2170	. . . . 5 ⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
58		imaeq2 4949	. . . . . 6 ⊢ (𝑧 = (𝑟 × 𝑠) → (^◡𝐻 “ 𝑧) = (^◡𝐻 “ (𝑟 × 𝑠)))
59	58	eleq1d 2239	. . . . 5 ⊢ (𝑧 = (𝑟 × 𝑠) → ((^◡𝐻 “ 𝑧) ∈ 𝑈 ↔ (^◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈))
60	57, 59	ralrnmpo 5967	. . . 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 133	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑧 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(^◡𝐻 “ 𝑧) ∈ 𝑈)
63	1	toptopon 12810	. . . 4 ⊢ (𝑈 ∈ Top ↔ 𝑈 ∈ (TopOn‘𝑊))
64	43, 63	sylib 121	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ (TopOn‘𝑊))
65		cntop2 12996	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
66		cntop2 12996	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
67		eqid 2170	. . . . 5 ⊢ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
68	67	txval 13049	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
69	65, 66, 68	syl2an 287	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×_t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
70		toptopon2 12811	. . . . 5 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
71	65, 70	sylib 121	. . . 4 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ (TopOn‘∪ 𝑅))
72		toptopon2 12811	. . . . 5 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
73	66, 72	sylib 121	. . . 4 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ (TopOn‘∪ 𝑆))
74		txtopon 13056	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
75	71, 73, 74	syl2an 287	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
76	64, 69, 75	tgcn 13002	. 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 939	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 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 ∀wral 2448 {crab 2452 Vcvv 2730 ∩ cin 3120 ⊆ wss 3121 ⟨cop 3586 ∪ cuni 3796 ↦ cmpt 4050 × cxp 4609 ^◡ccnv 4610 dom cdm 4611 ran crn 4612 “ cima 4614 Fn wfn 5193 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 ∈ cmpo 5855 topGenctg 12594 Topctop 12789 TopOnctopon 12802 Cn ccn 12979 ×_t ctx 13046

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-topgen 12600 df-top 12790 df-topon 12803 df-bases 12835 df-cn 12982 df-tx 13047

This theorem is referenced by: uptx 13068 cnmpt1t 13079 cnmpt2t 13087

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