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Theorem txdis1cn 21851
 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 (𝜑𝑋𝑉)
txdis1cn.j (𝜑𝐽 ∈ (TopOn‘𝑌))
txdis1cn.k (𝜑𝐾 ∈ Top)
txdis1cn.f (𝜑𝐹 Fn (𝑋 × 𝑌))
txdis1cn.1 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
Assertion
Ref Expression
txdis1cn (𝜑𝐹 ∈ ((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.
StepHypRef Expression
1 txdis1cn.f . . 3 (𝜑𝐹 Fn (𝑋 × 𝑌))
2 txdis1cn.j . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑌))
32adantr 474 . . . . . 6 ((𝜑𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑌))
4 txdis1cn.k . . . . . . . 8 (𝜑𝐾 ∈ Top)
5 eqid 2778 . . . . . . . . 9 𝐾 = 𝐾
65toptopon 21133 . . . . . . . 8 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
74, 6sylib 210 . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
87adantr 474 . . . . . 6 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘ 𝐾))
9 txdis1cn.1 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
10 cnf2 21465 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾)) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)):𝑌 𝐾)
113, 8, 9, 10syl3anc 1439 . . . . 5 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)):𝑌 𝐾)
12 eqid 2778 . . . . . 6 (𝑦𝑌 ↦ (𝑥𝐹𝑦)) = (𝑦𝑌 ↦ (𝑥𝐹𝑦))
1312fmpt 6646 . . . . 5 (∀𝑦𝑌 (𝑥𝐹𝑦) ∈ 𝐾 ↔ (𝑦𝑌 ↦ (𝑥𝐹𝑦)):𝑌 𝐾)
1411, 13sylibr 226 . . . 4 ((𝜑𝑥𝑋) → ∀𝑦𝑌 (𝑥𝐹𝑦) ∈ 𝐾)
1514ralrimiva 3148 . . 3 (𝜑 → ∀𝑥𝑋𝑦𝑌 (𝑥𝐹𝑦) ∈ 𝐾)
16 ffnov 7043 . . 3 (𝐹:(𝑋 × 𝑌)⟶ 𝐾 ↔ (𝐹 Fn (𝑋 × 𝑌) ∧ ∀𝑥𝑋𝑦𝑌 (𝑥𝐹𝑦) ∈ 𝐾))
171, 15, 16sylanbrc 578 . 2 (𝜑𝐹:(𝑋 × 𝑌)⟶ 𝐾)
18 cnvimass 5741 . . . . . . . 8 (𝐹𝑢) ⊆ dom 𝐹
191adantr 474 . . . . . . . . 9 ((𝜑𝑢𝐾) → 𝐹 Fn (𝑋 × 𝑌))
20 fndm 6237 . . . . . . . . 9 (𝐹 Fn (𝑋 × 𝑌) → dom 𝐹 = (𝑋 × 𝑌))
2119, 20syl 17 . . . . . . . 8 ((𝜑𝑢𝐾) → dom 𝐹 = (𝑋 × 𝑌))
2218, 21syl5sseq 3872 . . . . . . 7 ((𝜑𝑢𝐾) → (𝐹𝑢) ⊆ (𝑋 × 𝑌))
23 relxp 5375 . . . . . . 7 Rel (𝑋 × 𝑌)
24 relss 5456 . . . . . . 7 ((𝐹𝑢) ⊆ (𝑋 × 𝑌) → (Rel (𝑋 × 𝑌) → Rel (𝐹𝑢)))
2522, 23, 24mpisyl 21 . . . . . 6 ((𝜑𝑢𝐾) → Rel (𝐹𝑢))
26 elpreima 6602 . . . . . . . 8 (𝐹 Fn (𝑋 × 𝑌) → (⟨𝑥, 𝑧⟩ ∈ (𝐹𝑢) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢)))
2719, 26syl 17 . . . . . . 7 ((𝜑𝑢𝐾) → (⟨𝑥, 𝑧⟩ ∈ (𝐹𝑢) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢)))
28 opelxp 5393 . . . . . . . . 9 (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ↔ (𝑥𝑋𝑧𝑌))
29 df-ov 6927 . . . . . . . . . . 11 (𝑥𝐹𝑧) = (𝐹‘⟨𝑥, 𝑧⟩)
3029eqcomi 2787 . . . . . . . . . 10 (𝐹‘⟨𝑥, 𝑧⟩) = (𝑥𝐹𝑧)
3130eleq1i 2850 . . . . . . . . 9 ((𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢 ↔ (𝑥𝐹𝑧) ∈ 𝑢)
3228, 31anbi12i 620 . . . . . . . 8 ((⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢) ↔ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢))
33 simprll 769 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑥𝑋)
34 snelpwi 5146 . . . . . . . . . . . 12 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
3533, 34syl 17 . . . . . . . . . . 11 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑥} ∈ 𝒫 𝑋)
3612mptpreima 5884 . . . . . . . . . . . 12 ((𝑦𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}
379adantrr 707 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑋𝑧𝑌)) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
3837ad2ant2r 737 . . . . . . . . . . . . 13 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
39 simplr 759 . . . . . . . . . . . . 13 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑢𝐾)
40 cnima 21481 . . . . . . . . . . . . 13 (((𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾) ∧ 𝑢𝐾) → ((𝑦𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) ∈ 𝐽)
4138, 39, 40syl2anc 579 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ((𝑦𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) ∈ 𝐽)
4236, 41syl5eqelr 2864 . . . . . . . . . . 11 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ∈ 𝐽)
43 simprlr 770 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑧𝑌)
44 simprr 763 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (𝑥𝐹𝑧) ∈ 𝑢)
45 vsnid 4431 . . . . . . . . . . . . . 14 𝑥 ∈ {𝑥}
46 opelxp 5393 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑥 ∈ {𝑥} ∧ 𝑧 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
4745, 46mpbiran 699 . . . . . . . . . . . . 13 (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ 𝑧 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})
48 oveq2 6932 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑥𝐹𝑦) = (𝑥𝐹𝑧))
4948eleq1d 2844 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ((𝑥𝐹𝑦) ∈ 𝑢 ↔ (𝑥𝐹𝑧) ∈ 𝑢))
5049elrab 3572 . . . . . . . . . . . . 13 (𝑧 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ↔ (𝑧𝑌 ∧ (𝑥𝐹𝑧) ∈ 𝑢))
5147, 50bitri 267 . . . . . . . . . . . 12 (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑧𝑌 ∧ (𝑥𝐹𝑧) ∈ 𝑢))
5243, 44, 51sylanbrc 578 . . . . . . . . . . 11 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
53 relxp 5375 . . . . . . . . . . . . 13 Rel ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})
5453a1i 11 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → Rel ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
55 opelxp 5393 . . . . . . . . . . . . 13 (⟨𝑛, 𝑚⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
5633snssd 4573 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑥} ⊆ 𝑋)
5756sselda 3821 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ 𝑛 ∈ {𝑥}) → 𝑛𝑋)
5857adantrr 707 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑛𝑋)
59 elrabi 3567 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → 𝑚𝑌)
6059ad2antll 719 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑚𝑌)
61 opelxp 5393 . . . . . . . . . . . . . . . 16 (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ↔ (𝑛𝑋𝑚𝑌))
6258, 60, 61sylanbrc 578 . . . . . . . . . . . . . . 15 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌))
63 df-ov 6927 . . . . . . . . . . . . . . . . 17 (𝑛𝐹𝑚) = (𝐹‘⟨𝑛, 𝑚⟩)
64 elsni 4415 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ {𝑥} → 𝑛 = 𝑥)
6564ad2antrl 718 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑛 = 𝑥)
6665oveq1d 6939 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝑛𝐹𝑚) = (𝑥𝐹𝑚))
6763, 66syl5eqr 2828 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝐹‘⟨𝑛, 𝑚⟩) = (𝑥𝐹𝑚))
68 oveq2 6932 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑚 → (𝑥𝐹𝑦) = (𝑥𝐹𝑚))
6968eleq1d 2844 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑚 → ((𝑥𝐹𝑦) ∈ 𝑢 ↔ (𝑥𝐹𝑚) ∈ 𝑢))
7069elrab 3572 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ↔ (𝑚𝑌 ∧ (𝑥𝐹𝑚) ∈ 𝑢))
7170simprbi 492 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (𝑥𝐹𝑚) ∈ 𝑢)
7271ad2antll 719 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝑥𝐹𝑚) ∈ 𝑢)
7367, 72eqeltrd 2859 . . . . . . . . . . . . . . 15 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)
74 elpreima 6602 . . . . . . . . . . . . . . . . 17 (𝐹 Fn (𝑋 × 𝑌) → (⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
751, 74syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
7675ad3antrrr 720 . . . . . . . . . . . . . . 15 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
7762, 73, 76mpbir2and 703 . . . . . . . . . . . . . 14 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → ⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢))
7877ex 403 . . . . . . . . . . . . 13 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ((𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) → ⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢)))
7955, 78syl5bi 234 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (⟨𝑛, 𝑚⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) → ⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢)))
8054, 79relssdv 5461 . . . . . . . . . . 11 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (𝐹𝑢))
81 xpeq1 5371 . . . . . . . . . . . . . 14 (𝑎 = {𝑥} → (𝑎 × 𝑏) = ({𝑥} × 𝑏))
8281eleq2d 2845 . . . . . . . . . . . . 13 (𝑎 = {𝑥} → (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏)))
8381sseq1d 3851 . . . . . . . . . . . . 13 (𝑎 = {𝑥} → ((𝑎 × 𝑏) ⊆ (𝐹𝑢) ↔ ({𝑥} × 𝑏) ⊆ (𝐹𝑢)))
8482, 83anbi12d 624 . . . . . . . . . . . 12 (𝑎 = {𝑥} → ((⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ∧ ({𝑥} × 𝑏) ⊆ (𝐹𝑢))))
85 xpeq2 5378 . . . . . . . . . . . . . 14 (𝑏 = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → ({𝑥} × 𝑏) = ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
8685eleq2d 2845 . . . . . . . . . . . . 13 (𝑏 = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})))
8785sseq1d 3851 . . . . . . . . . . . . 13 (𝑏 = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (({𝑥} × 𝑏) ⊆ (𝐹𝑢) ↔ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (𝐹𝑢)))
8886, 87anbi12d 624 . . . . . . . . . . . 12 (𝑏 = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → ((⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ∧ ({𝑥} × 𝑏) ⊆ (𝐹𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ∧ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (𝐹𝑢))))
8984, 88rspc2ev 3526 . . . . . . . . . . 11 (({𝑥} ∈ 𝒫 𝑋 ∧ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ∈ 𝐽 ∧ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ∧ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (𝐹𝑢))) → ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)))
9035, 42, 52, 80, 89syl112anc 1442 . . . . . . . . . 10 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)))
91 opex 5166 . . . . . . . . . . 11 𝑥, 𝑧⟩ ∈ V
92 eleq1 2847 . . . . . . . . . . . . 13 (𝑣 = ⟨𝑥, 𝑧⟩ → (𝑣 ∈ (𝑎 × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏)))
9392anbi1d 623 . . . . . . . . . . . 12 (𝑣 = ⟨𝑥, 𝑧⟩ → ((𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))))
94932rexbidv 3242 . . . . . . . . . . 11 (𝑣 = ⟨𝑥, 𝑧⟩ → (∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)) ↔ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))))
9591, 94elab 3558 . . . . . . . . . 10 (⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))} ↔ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)))
9690, 95sylibr 226 . . . . . . . . 9 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))})
9796ex 403 . . . . . . . 8 ((𝜑𝑢𝐾) → (((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))}))
9832, 97syl5bi 234 . . . . . . 7 ((𝜑𝑢𝐾) → ((⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))}))
9927, 98sylbid 232 . . . . . 6 ((𝜑𝑢𝐾) → (⟨𝑥, 𝑧⟩ ∈ (𝐹𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))}))
10025, 99relssdv 5461 . . . . 5 ((𝜑𝑢𝐾) → (𝐹𝑢) ⊆ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))})
101 ssabral 3894 . . . . 5 ((𝐹𝑢) ⊆ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))} ↔ ∀𝑣 ∈ (𝐹𝑢)∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)))
102100, 101sylib 210 . . . 4 ((𝜑𝑢𝐾) → ∀𝑣 ∈ (𝐹𝑢)∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)))
103 txdis1cn.x . . . . . . 7 (𝜑𝑋𝑉)
104 distopon 21213 . . . . . . 7 (𝑋𝑉 → 𝒫 𝑋 ∈ (TopOn‘𝑋))
105103, 104syl 17 . . . . . 6 (𝜑 → 𝒫 𝑋 ∈ (TopOn‘𝑋))
106105adantr 474 . . . . 5 ((𝜑𝑢𝐾) → 𝒫 𝑋 ∈ (TopOn‘𝑋))
1072adantr 474 . . . . 5 ((𝜑𝑢𝐾) → 𝐽 ∈ (TopOn‘𝑌))
108 eltx 21784 . . . . 5 ((𝒫 𝑋 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑌)) → ((𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽) ↔ ∀𝑣 ∈ (𝐹𝑢)∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))))
109106, 107, 108syl2anc 579 . . . 4 ((𝜑𝑢𝐾) → ((𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽) ↔ ∀𝑣 ∈ (𝐹𝑢)∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))))
110102, 109mpbird 249 . . 3 ((𝜑𝑢𝐾) → (𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽))
111110ralrimiva 3148 . 2 (𝜑 → ∀𝑢𝐾 (𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽))
112 txtopon 21807 . . . 4 ((𝒫 𝑋 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑌)) → (𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)))
113105, 2, 112syl2anc 579 . . 3 (𝜑 → (𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)))
114 iscn 21451 . . 3 (((𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾) ↔ (𝐹:(𝑋 × 𝑌)⟶ 𝐾 ∧ ∀𝑢𝐾 (𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽))))
115113, 7, 114syl2anc 579 . 2 (𝜑 → (𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾) ↔ (𝐹:(𝑋 × 𝑌)⟶ 𝐾 ∧ ∀𝑢𝐾 (𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽))))
11617, 111, 115mpbir2and 703 1 (𝜑𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {cab 2763  ∀wral 3090  ∃wrex 3091  {crab 3094   ⊆ wss 3792  𝒫 cpw 4379  {csn 4398  ⟨cop 4404  ∪ cuni 4673   ↦ cmpt 4967   × cxp 5355  ◡ccnv 5356  dom cdm 5357   “ cima 5360  Rel wrel 5362   Fn wfn 6132  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924  Topctop 21109  TopOnctopon 21126   Cn ccn 21440   ×t ctx 21776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-map 8144  df-topgen 16494  df-top 21110  df-topon 21127  df-bases 21162  df-cn 21443  df-tx 21778 This theorem is referenced by:  tgpmulg2  22310
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