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Theorem txdis1cn 22219
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 484 . . . . . 6 ((𝜑𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑌))
4 txdis1cn.k . . . . . . . 8 (𝜑𝐾 ∈ Top)
5 toptopon2 21502 . . . . . . . 8 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
64, 5sylib 221 . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
76adantr 484 . . . . . 6 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘ 𝐾))
8 txdis1cn.1 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
9 cnf2 21833 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾)) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)):𝑌 𝐾)
103, 7, 8, 9syl3anc 1368 . . . . 5 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)):𝑌 𝐾)
11 eqid 2821 . . . . . 6 (𝑦𝑌 ↦ (𝑥𝐹𝑦)) = (𝑦𝑌 ↦ (𝑥𝐹𝑦))
1211fmpt 6847 . . . . 5 (∀𝑦𝑌 (𝑥𝐹𝑦) ∈ 𝐾 ↔ (𝑦𝑌 ↦ (𝑥𝐹𝑦)):𝑌 𝐾)
1310, 12sylibr 237 . . . 4 ((𝜑𝑥𝑋) → ∀𝑦𝑌 (𝑥𝐹𝑦) ∈ 𝐾)
1413ralrimiva 3170 . . 3 (𝜑 → ∀𝑥𝑋𝑦𝑌 (𝑥𝐹𝑦) ∈ 𝐾)
15 ffnov 7252 . . 3 (𝐹:(𝑋 × 𝑌)⟶ 𝐾 ↔ (𝐹 Fn (𝑋 × 𝑌) ∧ ∀𝑥𝑋𝑦𝑌 (𝑥𝐹𝑦) ∈ 𝐾))
161, 14, 15sylanbrc 586 . 2 (𝜑𝐹:(𝑋 × 𝑌)⟶ 𝐾)
17 cnvimass 5922 . . . . . . . 8 (𝐹𝑢) ⊆ dom 𝐹
181adantr 484 . . . . . . . . 9 ((𝜑𝑢𝐾) → 𝐹 Fn (𝑋 × 𝑌))
19 fndm 6428 . . . . . . . . 9 (𝐹 Fn (𝑋 × 𝑌) → dom 𝐹 = (𝑋 × 𝑌))
2018, 19syl 17 . . . . . . . 8 ((𝜑𝑢𝐾) → dom 𝐹 = (𝑋 × 𝑌))
2117, 20sseqtrid 3995 . . . . . . 7 ((𝜑𝑢𝐾) → (𝐹𝑢) ⊆ (𝑋 × 𝑌))
22 relxp 5546 . . . . . . 7 Rel (𝑋 × 𝑌)
23 relss 5629 . . . . . . 7 ((𝐹𝑢) ⊆ (𝑋 × 𝑌) → (Rel (𝑋 × 𝑌) → Rel (𝐹𝑢)))
2421, 22, 23mpisyl 21 . . . . . 6 ((𝜑𝑢𝐾) → Rel (𝐹𝑢))
25 elpreima 6801 . . . . . . . 8 (𝐹 Fn (𝑋 × 𝑌) → (⟨𝑥, 𝑧⟩ ∈ (𝐹𝑢) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢)))
2618, 25syl 17 . . . . . . 7 ((𝜑𝑢𝐾) → (⟨𝑥, 𝑧⟩ ∈ (𝐹𝑢) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢)))
27 opelxp 5564 . . . . . . . . 9 (⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ↔ (𝑥𝑋𝑧𝑌))
28 df-ov 7133 . . . . . . . . . . 11 (𝑥𝐹𝑧) = (𝐹‘⟨𝑥, 𝑧⟩)
2928eqcomi 2830 . . . . . . . . . 10 (𝐹‘⟨𝑥, 𝑧⟩) = (𝑥𝐹𝑧)
3029eleq1i 2902 . . . . . . . . 9 ((𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢 ↔ (𝑥𝐹𝑧) ∈ 𝑢)
3127, 30anbi12i 629 . . . . . . . 8 ((⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢) ↔ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢))
32 simprll 778 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑥𝑋)
33 snelpwi 5310 . . . . . . . . . . . 12 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
3432, 33syl 17 . . . . . . . . . . 11 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑥} ∈ 𝒫 𝑋)
3511mptpreima 6065 . . . . . . . . . . . 12 ((𝑦𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}
368adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑋𝑧𝑌)) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
3736ad2ant2r 746 . . . . . . . . . . . . 13 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))
38 simplr 768 . . . . . . . . . . . . 13 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑢𝐾)
39 cnima 21849 . . . . . . . . . . . . 13 (((𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾) ∧ 𝑢𝐾) → ((𝑦𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) ∈ 𝐽)
4037, 38, 39syl2anc 587 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ((𝑦𝑌 ↦ (𝑥𝐹𝑦)) “ 𝑢) ∈ 𝐽)
4135, 40eqeltrrid 2917 . . . . . . . . . . 11 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ∈ 𝐽)
42 simprlr 779 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → 𝑧𝑌)
43 simprr 772 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (𝑥𝐹𝑧) ∈ 𝑢)
44 vsnid 4575 . . . . . . . . . . . . . 14 𝑥 ∈ {𝑥}
45 opelxp 5564 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑥 ∈ {𝑥} ∧ 𝑧 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
4644, 45mpbiran 708 . . . . . . . . . . . . 13 (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ 𝑧 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})
47 oveq2 7138 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑥𝐹𝑦) = (𝑥𝐹𝑧))
4847eleq1d 2896 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ((𝑥𝐹𝑦) ∈ 𝑢 ↔ (𝑥𝐹𝑧) ∈ 𝑢))
4948elrab 3657 . . . . . . . . . . . . 13 (𝑧 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ↔ (𝑧𝑌 ∧ (𝑥𝐹𝑧) ∈ 𝑢))
5046, 49bitri 278 . . . . . . . . . . . 12 (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑧𝑌 ∧ (𝑥𝐹𝑧) ∈ 𝑢))
5142, 43, 50sylanbrc 586 . . . . . . . . . . 11 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
52 relxp 5546 . . . . . . . . . . . . 13 Rel ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})
5352a1i 11 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → Rel ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
54 opelxp 5564 . . . . . . . . . . . . 13 (⟨𝑛, 𝑚⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ↔ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
5532snssd 4715 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → {𝑥} ⊆ 𝑋)
5655sselda 3943 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ 𝑛 ∈ {𝑥}) → 𝑛𝑋)
5756adantrr 716 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑛𝑋)
58 elrabi 3652 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → 𝑚𝑌)
5958ad2antll 728 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑚𝑌)
6057, 59opelxpd 5566 . . . . . . . . . . . . . . 15 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌))
61 df-ov 7133 . . . . . . . . . . . . . . . . 17 (𝑛𝐹𝑚) = (𝐹‘⟨𝑛, 𝑚⟩)
62 elsni 4557 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ {𝑥} → 𝑛 = 𝑥)
6362ad2antrl 727 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → 𝑛 = 𝑥)
6463oveq1d 7145 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝑛𝐹𝑚) = (𝑥𝐹𝑚))
6561, 64syl5eqr 2870 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝐹‘⟨𝑛, 𝑚⟩) = (𝑥𝐹𝑚))
66 oveq2 7138 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑚 → (𝑥𝐹𝑦) = (𝑥𝐹𝑚))
6766eleq1d 2896 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑚 → ((𝑥𝐹𝑦) ∈ 𝑢 ↔ (𝑥𝐹𝑚) ∈ 𝑢))
6867elrab 3657 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} ↔ (𝑚𝑌 ∧ (𝑥𝐹𝑚) ∈ 𝑢))
6968simprbi 500 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (𝑥𝐹𝑚) ∈ 𝑢)
7069ad2antll 728 . . . . . . . . . . . . . . . 16 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝑥𝐹𝑚) ∈ 𝑢)
7165, 70eqeltrd 2912 . . . . . . . . . . . . . . 15 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)
72 elpreima 6801 . . . . . . . . . . . . . . . . 17 (𝐹 Fn (𝑋 × 𝑌) → (⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
731, 72syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
7473ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → (⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢) ↔ (⟨𝑛, 𝑚⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑛, 𝑚⟩) ∈ 𝑢)))
7560, 71, 74mpbir2and 712 . . . . . . . . . . . . . 14 ((((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) ∧ (𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})) → ⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢))
7675ex 416 . . . . . . . . . . . . 13 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ((𝑛 ∈ {𝑥} ∧ 𝑚 ∈ {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) → ⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢)))
7754, 76syl5bi 245 . . . . . . . . . . . 12 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → (⟨𝑛, 𝑚⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) → ⟨𝑛, 𝑚⟩ ∈ (𝐹𝑢)))
7853, 77relssdv 5634 . . . . . . . . . . 11 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (𝐹𝑢))
79 xpeq1 5542 . . . . . . . . . . . . . 14 (𝑎 = {𝑥} → (𝑎 × 𝑏) = ({𝑥} × 𝑏))
8079eleq2d 2897 . . . . . . . . . . . . 13 (𝑎 = {𝑥} → (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏)))
8179sseq1d 3974 . . . . . . . . . . . . 13 (𝑎 = {𝑥} → ((𝑎 × 𝑏) ⊆ (𝐹𝑢) ↔ ({𝑥} × 𝑏) ⊆ (𝐹𝑢)))
8280, 81anbi12d 633 . . . . . . . . . . . 12 (𝑎 = {𝑥} → ((⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ∧ ({𝑥} × 𝑏) ⊆ (𝐹𝑢))))
83 xpeq2 5549 . . . . . . . . . . . . . 14 (𝑏 = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → ({𝑥} × 𝑏) = ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}))
8483eleq2d 2897 . . . . . . . . . . . . 13 (𝑏 = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢})))
8583sseq1d 3974 . . . . . . . . . . . . 13 (𝑏 = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → (({𝑥} × 𝑏) ⊆ (𝐹𝑢) ↔ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (𝐹𝑢)))
8684, 85anbi12d 633 . . . . . . . . . . . 12 (𝑏 = {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢} → ((⟨𝑥, 𝑧⟩ ∈ ({𝑥} × 𝑏) ∧ ({𝑥} × 𝑏) ⊆ (𝐹𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ∧ ({𝑥} × {𝑦𝑌 ∣ (𝑥𝐹𝑦) ∈ 𝑢}) ⊆ (𝐹𝑢))))
8782, 86rspc2ev 3612 . . . . . . . . . . 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)))
8834, 41, 51, 78, 87syl112anc 1371 . . . . . . . . . 10 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)))
89 opex 5329 . . . . . . . . . . 11 𝑥, 𝑧⟩ ∈ V
90 eleq1 2899 . . . . . . . . . . . . 13 (𝑣 = ⟨𝑥, 𝑧⟩ → (𝑣 ∈ (𝑎 × 𝑏) ↔ ⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏)))
9190anbi1d 632 . . . . . . . . . . . 12 (𝑣 = ⟨𝑥, 𝑧⟩ → ((𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)) ↔ (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))))
92912rexbidv 3286 . . . . . . . . . . 11 (𝑣 = ⟨𝑥, 𝑧⟩ → (∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)) ↔ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))))
9389, 92elab 3644 . . . . . . . . . 10 (⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))} ↔ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (⟨𝑥, 𝑧⟩ ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)))
9488, 93sylibr 237 . . . . . . . . 9 (((𝜑𝑢𝐾) ∧ ((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢)) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))})
9594ex 416 . . . . . . . 8 ((𝜑𝑢𝐾) → (((𝑥𝑋𝑧𝑌) ∧ (𝑥𝐹𝑧) ∈ 𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))}))
9631, 95syl5bi 245 . . . . . . 7 ((𝜑𝑢𝐾) → ((⟨𝑥, 𝑧⟩ ∈ (𝑋 × 𝑌) ∧ (𝐹‘⟨𝑥, 𝑧⟩) ∈ 𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))}))
9726, 96sylbid 243 . . . . . 6 ((𝜑𝑢𝐾) → (⟨𝑥, 𝑧⟩ ∈ (𝐹𝑢) → ⟨𝑥, 𝑧⟩ ∈ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))}))
9824, 97relssdv 5634 . . . . 5 ((𝜑𝑢𝐾) → (𝐹𝑢) ⊆ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))})
99 ssabral 4018 . . . . 5 ((𝐹𝑢) ⊆ {𝑣 ∣ ∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))} ↔ ∀𝑣 ∈ (𝐹𝑢)∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)))
10098, 99sylib 221 . . . 4 ((𝜑𝑢𝐾) → ∀𝑣 ∈ (𝐹𝑢)∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢)))
101 txdis1cn.x . . . . . . 7 (𝜑𝑋𝑉)
102 distopon 21581 . . . . . . 7 (𝑋𝑉 → 𝒫 𝑋 ∈ (TopOn‘𝑋))
103101, 102syl 17 . . . . . 6 (𝜑 → 𝒫 𝑋 ∈ (TopOn‘𝑋))
104103adantr 484 . . . . 5 ((𝜑𝑢𝐾) → 𝒫 𝑋 ∈ (TopOn‘𝑋))
1052adantr 484 . . . . 5 ((𝜑𝑢𝐾) → 𝐽 ∈ (TopOn‘𝑌))
106 eltx 22152 . . . . 5 ((𝒫 𝑋 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑌)) → ((𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽) ↔ ∀𝑣 ∈ (𝐹𝑢)∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))))
107104, 105, 106syl2anc 587 . . . 4 ((𝜑𝑢𝐾) → ((𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽) ↔ ∀𝑣 ∈ (𝐹𝑢)∃𝑎 ∈ 𝒫 𝑋𝑏𝐽 (𝑣 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐹𝑢))))
108100, 107mpbird 260 . . 3 ((𝜑𝑢𝐾) → (𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽))
109108ralrimiva 3170 . 2 (𝜑 → ∀𝑢𝐾 (𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽))
110 txtopon 22175 . . . 4 ((𝒫 𝑋 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑌)) → (𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)))
111103, 2, 110syl2anc 587 . . 3 (𝜑 → (𝒫 𝑋 ×t 𝐽) ∈ (TopOn‘(𝑋 × 𝑌)))
112 iscn 21819 . . 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))))
113111, 6, 112syl2anc 587 . 2 (𝜑 → (𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾) ↔ (𝐹:(𝑋 × 𝑌)⟶ 𝐾 ∧ ∀𝑢𝐾 (𝐹𝑢) ∈ (𝒫 𝑋 ×t 𝐽))))
11416, 109, 113mpbir2and 712 1 (𝜑𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2799  wral 3126  wrex 3127  {crab 3130  wss 3910  𝒫 cpw 4512  {csn 4540  cop 4546   cuni 4811  cmpt 5119   × cxp 5526  ccnv 5527  dom cdm 5528  cima 5531  Rel wrel 5533   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7130  Topctop 21477  TopOnctopon 21494   Cn ccn 21808   ×t ctx 22144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-map 8383  df-topgen 16696  df-top 21478  df-topon 21495  df-bases 21530  df-cn 21811  df-tx 22146
This theorem is referenced by:  tgpmulg2  22678
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