Theorem txdis1cn 13817

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 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑌))
4		txdis1cn.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Top)
5		toptopon2 13558	. . . . . . . 8 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
6	4, 5	sylib 122	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
7	6	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
8		txdis1cn.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
9		cnf2 13744	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)):𝑌⟶∪ 𝐾)
10	3, 7, 8, 9	syl3anc 1238	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)):𝑌⟶∪ 𝐾)
11		eqid 2177	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦))
12	11	fmpt 5668	. . . . 5 ⊢ (∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) ∈ ∪ 𝐾 ↔ (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)):𝑌⟶∪ 𝐾)
13	10, 12	sylibr 134	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) ∈ ∪ 𝐾)
14	13	ralrimiva 2550	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) ∈ ∪ 𝐾)
15		ffnov 5981	. . 3 ⊢ (𝐹:(𝑋 × 𝑌)⟶∪ 𝐾 ↔ (𝐹 Fn (𝑋 × 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) ∈ ∪ 𝐾))
16	1, 14, 15	sylanbrc 417	. 2 ⊢ (𝜑 → 𝐹:(𝑋 × 𝑌)⟶∪ 𝐾)
17		cnvimass 4993	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑢) ⊆ dom 𝐹
18	1	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → 𝐹 Fn (𝑋 × 𝑌))
19		fndm 5317	. . . . . . . . 9 ⊢ (𝐹 Fn (𝑋 × 𝑌) → dom 𝐹 = (𝑋 × 𝑌))
20	18, 19	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → dom 𝐹 = (𝑋 × 𝑌))
21	17, 20	sseqtrid 3207	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (◡𝐹 “ 𝑢) ⊆ (𝑋 × 𝑌))
22		relxp 4737	. . . . . . 7 ⊢ Rel (𝑋 × 𝑌)
23		relss 4715	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑢) ⊆ (𝑋 × 𝑌) → (Rel (𝑋 × 𝑌) → Rel (◡𝐹 “ 𝑢)))
24	21, 22, 23	mpisyl 1446	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → Rel (◡𝐹 “ 𝑢))
25		elpreima 5637	. . . . . . . 8 ⊢ (𝐹 Fn (𝑋 × 𝑌) → (⟨𝑥, 𝑧⟩ ∈ (◡𝐹 “ 𝑢) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢)))
26	18, 25	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (⟨𝑥, 𝑧⟩ ∈ (◡𝐹 “ 𝑢) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢)))
27		opelxp 4658	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌))
28		df-ov 5880	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑧) = (𝐹‘⟨𝑥, 𝑧⟩)
29	28	eqcomi 2181	. . . . . . . . . 10 ⊢ (𝐹‘⟨𝑥, 𝑧⟩) = (𝑥𝐹𝑧)
30	29	eleq1i 2243	. . . . . . . . 9 ⊢ ((𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢 ↔ (𝑥𝐹𝑧) ∈ 𝑢)
31	27, 30	anbi12i 460	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢))
32		simprll 537	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑥 ∈ 𝑋)
33		snelpwi 4214	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
34	32, 33	syl 14	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑥} ∈ 𝒫 𝑋)
35	11	mptpreima 5124	. . . . . . . . . . . 12 ⊢ (◡(𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) = {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}
36	8	adantrr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
37	36	ad2ant2r 509	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
38		simplr 528	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑢 ∈ 𝐾)
39		cnima 13759	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾) ∧ 𝑢 ∈ 𝐾) → (◡(𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) ∈ 𝐽)
40	37, 38, 39	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (◡(𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) ∈ 𝐽)
41	35, 40	eqeltrrid 2265	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ∈ 𝐽)
42		simprlr 538	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑧 ∈ 𝑌)
43		simprr 531	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (𝑥𝐹𝑧) ∈ 𝑢)
44		vsnid 3626	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ {𝑥}
45		opelxp 4658	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑥 ∈ {𝑥} ∧ 𝑧 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
46	44, 45	mpbiran 940	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ 𝑧 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})
47		oveq2 5885	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦) = (𝑥𝐹𝑧))
48	47	eleq1d 2246	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ((𝑥𝐹𝑦) ∈ 𝑢 ↔ (𝑥𝐹𝑧) ∈ 𝑢))
49	48	elrab 2895	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ↔ (𝑧 ∈ 𝑌 ∧ (𝑥𝐹𝑧) ∈ 𝑢))
50	46, 49	bitri 184	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑧 ∈ 𝑌 ∧ (𝑥𝐹𝑧) ∈ 𝑢))
51	42, 43, 50	sylanbrc 417	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
52		relxp 4737	. . . . . . . . . . . . 13 ⊢ Rel ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})
53	52	a1i 9	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → Rel ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
54		opelxp 4658	. . . . . . . . . . . . 13 ⊢ (⟨𝑛, 𝑚⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
55	32	snssd 3739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑥} ⊆ 𝑋)
56	55	sselda 3157	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ 𝑛 ∈ {𝑥}) → 𝑛 ∈ 𝑋)
57	56	adantrr 479	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑛 ∈ 𝑋)
58		elrabi 2892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → 𝑚 ∈ 𝑌)
59	58	ad2antll 491	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑚 ∈ 𝑌)
60	57, 59	opelxpd 4661	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌))
61		df-ov 5880	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛𝐹𝑚) = (𝐹‘⟨𝑛, 𝑚⟩)
62		elsni 3612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ {𝑥} → 𝑛 = 𝑥)
63	62	ad2antrl 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑛 = 𝑥)
64	63	oveq1d 5892	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝑛𝐹𝑚) = (𝑥𝐹𝑚))
65	61, 64	eqtr3id 2224	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝐹‘⟨𝑛, 𝑚⟩) = (𝑥𝐹𝑚))
66		oveq2 5885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑚 → (𝑥𝐹𝑦) = (𝑥𝐹𝑚))
67	66	eleq1d 2246	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑚 → ((𝑥𝐹𝑦) ∈ 𝑢 ↔ (𝑥𝐹𝑚) ∈ 𝑢))
68	67	elrab 2895	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ↔ (𝑚 ∈ 𝑌 ∧ (𝑥𝐹𝑚) ∈ 𝑢))
69	68	simprbi 275	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (𝑥𝐹𝑚) ∈ 𝑢)
70	69	ad2antll 491	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝑥𝐹𝑚) ∈ 𝑢)
71	65, 70	eqeltrd 2254	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)
72		elpreima 5637	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (𝑋 × 𝑌) → (⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
73	1, 72	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
74	73	ad3antrrr 492	. . . . . . . . . . . . . . 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)))
75	60, 71, 74	mpbir2and 944	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → ⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢))
76	75	ex 115	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ((𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) → ⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢)))
77	54, 76	biimtrid 152	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (⟨𝑛, 𝑚⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) → ⟨𝑛, 𝑚⟩ ∈ (◡𝐹 “ 𝑢)))
78	53, 77	relssdv 4720	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (◡𝐹 “ 𝑢))
79		xpeq1 4642	. . . . . . . . . . . . . 14 ⊢ (𝑎 = {𝑥} → (𝑎 × 𝑏) = ({𝑥} × 𝑏))
80	79	eleq2d 2247	. . . . . . . . . . . . 13 ⊢ (𝑎 = {𝑥} → (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏)))
81	79	sseq1d 3186	. . . . . . . . . . . . 13 ⊢ (𝑎 = {𝑥} → ((𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢) ↔ ({𝑥} × 𝑏) ⊆ (◡𝐹 “ 𝑢)))
82	80, 81	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑎 = {𝑥} → ((⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ∧ ({𝑥} × 𝑏) ⊆ (◡𝐹 “ 𝑢))))
83		xpeq2 4643	. . . . . . . . . . . . . 14 ⊢ (𝑏 = {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → ({𝑥} × 𝑏) = ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
84	83	eleq2d 2247	. . . . . . . . . . . . 13 ⊢ (𝑏 = {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})))
85	83	sseq1d 3186	. . . . . . . . . . . . 13 ⊢ (𝑏 = {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (({𝑥} × 𝑏) ⊆ (◡𝐹 “ 𝑢) ↔ ({𝑥} × {𝑦 ∈ 𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (◡𝐹 “ 𝑢)))
86	84, 85	anbi12d 473	. . . . . . . . . . . 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))))
87	82, 86	rspc2ev 2858	. . . . . . . . . . 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)))
88	34, 41, 51, 78, 87	syl112anc 1242	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐾) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)))
89		vex 2742	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
90		vex 2742	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
91	89, 90	opex 4231	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑧⟩ ∈ V
92		eleq1 2240	. . . . . . . . . . . . 13 ⊢ (𝑣 = ⟨𝑥, 𝑧⟩ → (𝑣 ∈ (𝑎 × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏)))
93	92	anbi1d 465	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨𝑥, 𝑧⟩ → ((𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))))
94	93	2rexbidv 2502	. . . . . . . . . . 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 2883	. . . . . . . . . 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	88, 95	sylibr 134	. . . . . . . . 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 115	. . . . . . . 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	31, 97	biimtrid 152	. . . . . . 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	26, 98	sylbid 150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (⟨𝑥, 𝑧⟩ ∈ (◡𝐹 “ 𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))}))
100	24, 99	relssdv 4720	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (◡𝐹 “ 𝑢) ⊆ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋∃𝑏 ∈ 𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (◡𝐹 “ 𝑢))})
101		ssabral 3228	. . . . 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 122	. . . 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 13626	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ (TopOn‘𝑋))
105	103, 104	syl 14	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ (TopOn‘𝑋))
106	105	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → 𝒫 𝑋 ∈ (TopOn‘𝑋))
107	2	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → 𝐽 ∈ (TopOn‘𝑌))
108		eltx 13798	. . . . 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 411	. . . 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 167	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → (◡𝐹 “ 𝑢) ∈ (𝒫 𝑋 ×t 𝐽))
111	110	ralrimiva 2550	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝐾 (◡𝐹 “ 𝑢) ∈ (𝒫 𝑋 ×t 𝐽))
112		txtopon 13801	. . . 4 ⊢ ((𝒫 𝑋 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑌)) → (𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)))
113	105, 2, 112	syl2anc 411	. . 3 ⊢ (𝜑 → (𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)))
114		iscn 13736	. . 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, 6, 114	syl2anc 411	. 2 ⊢ (𝜑 → (𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾) ↔ (𝐹:(𝑋 × 𝑌)⟶∪ 𝐾 ∧ ∀𝑢 ∈ 𝐾 (◡𝐹 “ 𝑢) ∈ (𝒫 𝑋 ×t 𝐽))))
116	16, 111, 115	mpbir2and 944	1 ⊢ (𝜑 → 𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  {crab 2459   ⊆ wss 3131  𝒫 cpw 3577  {csn 3594  ⟨cop 3597  ∪ cuni 3811   ↦ cmpt 4066   × cxp 4626  ^◡ccnv 4627  dom cdm 4628   “ cima 4631  Rel wrel 4633   Fn wfn 5213  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877  Topctop 13536  TopOnctopon 13549   Cn ccn 13724   ×_t ctx 13791

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-topgen 12714  df-top 13537  df-topon 13550  df-bases 13582  df-cn 13727  df-tx 13792

This theorem is referenced by: (None)