Theorem txdis1cn 21248

Description: A function is jointly continuous on a discrete left topology iff it is continuous as a function of its right argument, for each fixed left value. (Contributed by Mario Carneiro, 19-Sep-2015.)

Hypotheses

Ref	Expression
txdis1cn.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
txdis1cn.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑌))
txdis1cn.k	⊢ (𝜑 → 𝐾 ∈ Top)
txdis1cn.f	⊢ (𝜑 → 𝐹 Fn (𝑋 × 𝑌))
txdis1cn.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))

Assertion

Ref	Expression
txdis1cn	⊢ (𝜑 → 𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐽   𝑥,𝑋,𝑦   𝑥,𝐾,𝑦   𝜑,𝑥   𝑥,𝑌,𝑦

Allowed substitution hints:   𝜑(𝑦)   𝐽(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem txdis1cn

Dummy variables 𝑎 𝑏 𝑚 𝑛 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txdis1cn.f	. . 3 ⊢ (𝜑 → 𝐹 Fn (𝑋 × 𝑌))
2		txdis1cn.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑌))
3	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑌))
4		txdis1cn.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Top)
5		eqid 2610	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ 𝐾
6	5	toptopon 20548	. . . . . . . 8 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
7	4, 6	sylib 207	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
8	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
9		txdis1cn.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
10		cnf2 20863	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)):𝑌⟶∪ 𝐾)
11	3, 8, 9, 10	syl3anc 1318	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)):𝑌⟶∪ 𝐾)
12		eqid 2610	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦))
13	12	fmpt 6289	. . . . 5 ⊢ (∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) ∈ ∪ 𝐾 ↔ (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)):𝑌⟶∪ 𝐾)
14	11, 13	sylibr 223	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) ∈ ∪ 𝐾)
15	14	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) ∈ ∪ 𝐾)
16		ffnov 6662	. . 3 ⊢ (𝐹:(𝑋 × 𝑌)⟶∪ 𝐾 ↔ (𝐹 Fn (𝑋 × 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) ∈ ∪ 𝐾))
17	1, 15, 16	sylanbrc 695	. 2 ⊢ (𝜑 → 𝐹:(𝑋 × 𝑌)⟶∪ 𝐾)
18		cnvimass 5404	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑢) ⊆ dom 𝐹
19	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → 𝐹 Fn (𝑋 × 𝑌))
20		fndm 5904	. . . . . . . . 9 ⊢ (𝐹 Fn (𝑋 × 𝑌) → dom 𝐹 = (𝑋 × 𝑌))
21	19, 20	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → dom 𝐹 = (𝑋 × 𝑌))
22	18, 21	syl5sseq 3616	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (◡𝐹 “ 𝑢) ⊆ (𝑋 × 𝑌))
23		relxp 5150	. . . . . . 7 ⊢ Rel (𝑋 × 𝑌)
24		relss 5129	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑢) ⊆ (𝑋 × 𝑌) → (Rel (𝑋 × 𝑌) → Rel (◡𝐹 “ 𝑢)))
25	22, 23, 24	mpisyl 21	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → Rel (◡𝐹 “ 𝑢))
26		elpreima 6245	. . . . . . . 8 ⊢ (𝐹 Fn (𝑋 × 𝑌) → (⟨𝑥, 𝑧⟩ ∈ (◡𝐹 “ 𝑢) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢)))
27	19, 26	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (⟨𝑥, 𝑧⟩ ∈ (◡𝐹 “ 𝑢) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢)))
28		opelxp 5070	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌))
29		df-ov 6552	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑧) = (𝐹‘⟨𝑥, 𝑧⟩)
30	29	eqcomi 2619	. . . . . . . . . 10 ⊢ (𝐹‘⟨𝑥, 𝑧⟩) = (𝑥𝐹𝑧)
31	30	eleq1i 2679	. . . . . . . . 9 ⊢ ((𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢 ↔ (𝑥𝐹𝑧) ∈ 𝑢)
32	28, 31	anbi12i 729	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢))
33		simprll 798	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑥 ∈ 𝑋)
34		snelpwi 4839	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
35	33, 34	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑥} ∈ 𝒫 𝑋)
36	12	mptpreima 5545	. . . . . . . . . . . 12 ⊢ (◡(𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) = {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}
37	9	adantrr 749	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
38	37	ad2ant2r 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
39		simplr 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑢 ∈ 𝐾)
40		cnima 20879	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾) ∧ 𝑢 ∈ 𝐾) → (◡(𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) ∈ 𝐽)
41	38, 39, 40	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (◡(𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) ∈ 𝐽)
42	36, 41	syl5eqelr 2693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ∈ 𝐽)
43		simprlr 799	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑧 ∈ 𝑌)
44		simprr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (𝑥𝐹𝑧) ∈ 𝑢)
45		vsnid 4156	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ {𝑥}
46		opelxp 5070	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑥 ∈ {𝑥} ∧ 𝑧 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
47	45, 46	mpbiran 955	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ 𝑧 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})
48		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦) = (𝑥𝐹𝑧))
49	48	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ((𝑥𝐹𝑦) ∈ 𝑢 ↔ (𝑥𝐹𝑧) ∈ 𝑢))
50	49	elrab 3331	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ↔ (𝑧 ∈ 𝑌 ∧ (𝑥𝐹𝑧) ∈ 𝑢))
51	47, 50	bitri 263	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑧 ∈ 𝑌 ∧ (𝑥𝐹𝑧) ∈ 𝑢))
52	43, 44, 51	sylanbrc 695	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
53		relxp 5150	. . . . . . . . . . . . 13 ⊢ Rel ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})
54	53	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → Rel ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
55		opelxp 5070	. . . . . . . . . . . . 13 ⊢ (⟨𝑛, 𝑚⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
56	33	snssd 4281	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑥} ⊆ 𝑋)
57	56	sselda 3568	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ 𝑛 ∈ {𝑥}) → 𝑛 ∈ 𝑋)
58	57	adantrr 749	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑛 ∈ 𝑋)
59		elrabi 3328	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → 𝑚 ∈ 𝑌)
60	59	ad2antll 761	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑚 ∈ 𝑌)
61		opelxp 5070	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ↔ (𝑛 ∈ 𝑋 ∧ 𝑚 ∈ 𝑌))
62	58, 60, 61	sylanbrc 695	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌))
63		df-ov 6552	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛𝐹𝑚) = (𝐹‘⟨𝑛, 𝑚⟩)
64		elsni 4142	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ {𝑥} → 𝑛 = 𝑥)
65	64	ad2antrl 760	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑛 = 𝑥)
66	65	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝑛𝐹𝑚) = (𝑥𝐹𝑚))
67	63, 66	syl5eqr 2658	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝐹‘⟨𝑛, 𝑚⟩) = (𝑥𝐹𝑚))
68		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑚 → (𝑥𝐹𝑦) = (𝑥𝐹𝑚))
69	68	eleq1d 2672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑚 → ((𝑥𝐹𝑦) ∈ 𝑢 ↔ (𝑥𝐹𝑚) ∈ 𝑢))
70	69	elrab 3331	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ↔ (𝑚 ∈ 𝑌 ∧ (𝑥𝐹𝑚) ∈ 𝑢))
71	70	simprbi 479	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (𝑥𝐹𝑚) ∈ 𝑢)
72	71	ad2antll 761	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝑥𝐹𝑚) ∈ 𝑢)
73	67, 72	eqeltrd 2688	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)
74		elpreima 6245	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (𝑋 × 𝑌) → (⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
75	1, 74	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
76	75	ad3antrrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
77	62, 73, 76	mpbir2and 959	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → ⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢))
78	77	ex 449	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ((𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) → ⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢)))
79	55, 78	syl5bi 231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (⟨𝑛, 𝑚⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) → ⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢)))
80	54, 79	relssdv 5135	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (◡𝐹 “ 𝑢))
81		xpeq1 5052	. . . . . . . . . . . . . 14 ⊢ (𝑎 = {𝑥} → (𝑎 × 𝑏) = ({𝑥} × 𝑏))
82	81	eleq2d 2673	. . . . . . . . . . . . 13 ⊢ (𝑎 = {𝑥} → (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏)))
83	81	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑎 = {𝑥} → ((𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢) ↔ ({𝑥} × 𝑏) ⊆ (◡𝐹 “ 𝑢)))
84	82, 83	anbi12d 743	. . . . . . . . . . . 12 ⊢ (𝑎 = {𝑥} → ((⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ∧ ({𝑥} × 𝑏) ⊆ (◡𝐹 “ 𝑢))))
85		xpeq2 5053	. . . . . . . . . . . . . 14 ⊢ (𝑏 = {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → ({𝑥} × 𝑏) = ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
86	85	eleq2d 2673	. . . . . . . . . . . . 13 ⊢ (𝑏 = {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})))
87	85	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑏 = {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (({𝑥} × 𝑏) ⊆ (◡𝐹 “ 𝑢) ↔ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (◡𝐹 “ 𝑢)))
88	86, 87	anbi12d 743	. . . . . . . . . . . 12 ⊢ (𝑏 = {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → ((⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ∧ ({𝑥} × 𝑏) ⊆ (◡𝐹 “ 𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ∧ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (◡𝐹 “ 𝑢))))
89	84, 88	rspc2ev 3295	. . . . . . . . . . 11 ⊢ (({𝑥} ∈ 𝒫 𝑋 ∧ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ∈ 𝐽 ∧ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ∧ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (◡𝐹 “ 𝑢))) → ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)))
90	35, 42, 52, 80, 89	syl112anc 1322	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)))
91		opex 4859	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑧⟩ ∈ V
92		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑣 = ⟨𝑥, 𝑧⟩ → (𝑣 ∈ (𝑎 × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏)))
93	92	anbi1d 737	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨𝑥, 𝑧⟩ → ((𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))))
94	93	2rexbidv 3039	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑥, 𝑧⟩ → (∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)) ↔ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))))
95	91, 94	elab 3319	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))} ↔ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)))
96	90, 95	sylibr 223	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))})
97	96	ex 449	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))}))
98	32, 97	syl5bi 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → ((⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))}))
99	27, 98	sylbid 229	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (⟨𝑥, 𝑧⟩ ∈ (◡𝐹 “ 𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))}))
100	25, 99	relssdv 5135	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (◡𝐹 “ 𝑢) ⊆ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))})
101		ssabral 3636	. . . . 5 ⊢ ((◡𝐹 “ 𝑢) ⊆ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))} ↔ ∀𝑣 ∈ (◡𝐹 “ 𝑢)∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)))
102	100, 101	sylib 207	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → ∀𝑣 ∈ (◡𝐹 “ 𝑢)∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)))
103		txdis1cn.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
104		distopon 20611	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ (TopOn‘𝑋))
105	103, 104	syl 17	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ (TopOn‘𝑋))
106	105	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → 𝒫 𝑋 ∈ (TopOn‘𝑋))
107	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → 𝐽 ∈ (TopOn‘𝑌))
108		eltx 21181	. . . . 5 ⊢ ((𝒫 𝑋 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑌)) → ((◡𝐹 “ 𝑢) ∈ (𝒫 𝑋 ×t 𝐽) ↔ ∀𝑣 ∈ (◡𝐹 “ 𝑢)∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))))
109	106, 107, 108	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → ((◡𝐹 “ 𝑢) ∈ (𝒫 𝑋 ×t 𝐽) ↔ ∀𝑣 ∈ (◡𝐹 “ 𝑢)∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))))
110	102, 109	mpbird 246	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (◡𝐹 “ 𝑢) ∈ (𝒫 𝑋 ×t 𝐽))
111	110	ralrimiva 2949	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝐾 (◡𝐹 “ 𝑢) ∈ (𝒫 𝑋 ×t 𝐽))
112		txtopon 21204	. . . 4 ⊢ ((𝒫 𝑋 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑌)) → (𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)))
113	105, 2, 112	syl2anc 691	. . 3 ⊢ (𝜑 → (𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)))
114		iscn 20849	. . 3 ⊢ (((𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾) ↔ (𝐹:(𝑋 × 𝑌)⟶∪ 𝐾 ∧ ∀𝑢 ∈ 𝐾 (◡𝐹 “ 𝑢) ∈ (𝒫 𝑋 ×t 𝐽))))
115	113, 7, 114	syl2anc 691	. 2 ⊢ (𝜑 → (𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾) ↔ (𝐹:(𝑋 × 𝑌)⟶∪ 𝐾 ∧ ∀𝑢 ∈ 𝐾 (◡𝐹 “ 𝑢) ∈ (𝒫 𝑋 ×t 𝐽))))
116	17, 111, 115	mpbir2and 959	1 ⊢ (𝜑 → 𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  dom cdm 5038   “ cima 5041  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Topctop 20517  TopOnctopon 20518   Cn ccn 20838   ×_t ctx 21173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-tx 21175

This theorem is referenced by:  tgpmulg2  21708