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Theorem txflf 22019
Description: Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypotheses
Ref Expression
txflf.j (𝜑𝐽 ∈ (TopOn‘𝑋))
txflf.k (𝜑𝐾 ∈ (TopOn‘𝑌))
txflf.l (𝜑𝐿 ∈ (Fil‘𝑍))
txflf.f (𝜑𝐹:𝑍𝑋)
txflf.g (𝜑𝐺:𝑍𝑌)
txflf.h 𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
Assertion
Ref Expression
txflf (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺))))
Distinct variable groups:   𝜑,𝑛   𝑛,𝐹   𝑛,𝐺   𝑛,𝑍   𝑛,𝑋   𝑛,𝑌
Allowed substitution hints:   𝑅(𝑛)   𝑆(𝑛)   𝐻(𝑛)   𝐽(𝑛)   𝐾(𝑛)   𝐿(𝑛)

Proof of Theorem txflf
Dummy variables 𝑢 𝑣 𝑧 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3394 . . . . . . . 8 𝑢 ∈ V
2 vex 3394 . . . . . . . 8 𝑣 ∈ V
31, 2xpex 7188 . . . . . . 7 (𝑢 × 𝑣) ∈ V
43rgen2w 3113 . . . . . 6 𝑢𝐽𝑣𝐾 (𝑢 × 𝑣) ∈ V
5 eqid 2806 . . . . . . 7 (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
6 eleq2 2874 . . . . . . . 8 (𝑧 = (𝑢 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑧 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣)))
7 sseq2 3824 . . . . . . . . 9 (𝑧 = (𝑢 × 𝑣) → ((𝐻) ⊆ 𝑧 ↔ (𝐻) ⊆ (𝑢 × 𝑣)))
87rexbidv 3240 . . . . . . . 8 (𝑧 = (𝑢 × 𝑣) → (∃𝐿 (𝐻) ⊆ 𝑧 ↔ ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)))
96, 8imbi12d 335 . . . . . . 7 (𝑧 = (𝑢 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣))))
105, 9ralrnmpt2 7001 . . . . . 6 (∀𝑢𝐽𝑣𝐾 (𝑢 × 𝑣) ∈ V → (∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧) ↔ ∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣))))
114, 10ax-mp 5 . . . . 5 (∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧) ↔ ∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)))
12 opelxp 5346 . . . . . . . . . . . . . . . 16 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑅𝑢𝑆𝑣))
13 ancom 450 . . . . . . . . . . . . . . . 16 ((𝑅𝑢𝑆𝑣) ↔ (𝑆𝑣𝑅𝑢))
1412, 13bitri 266 . . . . . . . . . . . . . . 15 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑆𝑣𝑅𝑢))
1514a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑆𝑣𝑅𝑢)))
16 r19.40 3276 . . . . . . . . . . . . . . . . 17 (∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) → (∃𝐿𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝐿𝑛 (𝐺𝑛) ∈ 𝑣))
17 raleq 3327 . . . . . . . . . . . . . . . . . . 19 ( = 𝑓 → (∀𝑛 (𝐹𝑛) ∈ 𝑢 ↔ ∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
1817cbvrexv 3361 . . . . . . . . . . . . . . . . . 18 (∃𝐿𝑛 (𝐹𝑛) ∈ 𝑢 ↔ ∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢)
19 raleq 3327 . . . . . . . . . . . . . . . . . . 19 ( = 𝑔 → (∀𝑛 (𝐺𝑛) ∈ 𝑣 ↔ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
2019cbvrexv 3361 . . . . . . . . . . . . . . . . . 18 (∃𝐿𝑛 (𝐺𝑛) ∈ 𝑣 ↔ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣)
2118, 20anbi12i 614 . . . . . . . . . . . . . . . . 17 ((∃𝐿𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝐿𝑛 (𝐺𝑛) ∈ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
2216, 21sylib 209 . . . . . . . . . . . . . . . 16 (∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) → (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
23 reeanv 3295 . . . . . . . . . . . . . . . . 17 (∃𝑓𝐿𝑔𝐿 (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
24 txflf.l . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐿 ∈ (Fil‘𝑍))
25 filin 21867 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑓𝐿𝑔𝐿) → (𝑓𝑔) ∈ 𝐿)
26253expb 1142 . . . . . . . . . . . . . . . . . . . . 21 ((𝐿 ∈ (Fil‘𝑍) ∧ (𝑓𝐿𝑔𝐿)) → (𝑓𝑔) ∈ 𝐿)
2724, 26sylan 571 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓𝐿𝑔𝐿)) → (𝑓𝑔) ∈ 𝐿)
28 inss1 4029 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑔) ⊆ 𝑓
29 ssralv 3863 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑔) ⊆ 𝑓 → (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 → ∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢))
3028, 29ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 → ∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢)
31 inss2 4030 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑔) ⊆ 𝑔
32 ssralv 3863 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑔) ⊆ 𝑔 → (∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣 → ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣))
3331, 32ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣 → ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣)
3430, 33anim12i 602 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → (∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣))
35 raleq 3327 . . . . . . . . . . . . . . . . . . . . . 22 ( = (𝑓𝑔) → (∀𝑛 (𝐹𝑛) ∈ 𝑢 ↔ ∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢))
36 raleq 3327 . . . . . . . . . . . . . . . . . . . . . 22 ( = (𝑓𝑔) → (∀𝑛 (𝐺𝑛) ∈ 𝑣 ↔ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣))
3735, 36anbi12d 618 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑓𝑔) → ((∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) ↔ (∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣)))
3837rspcev 3502 . . . . . . . . . . . . . . . . . . . 20 (((𝑓𝑔) ∈ 𝐿 ∧ (∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣)) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
3927, 34, 38syl2an 585 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓𝐿𝑔𝐿)) ∧ (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣)) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
4039ex 399 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑓𝐿𝑔𝐿)) → ((∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
4140rexlimdvva 3226 . . . . . . . . . . . . . . . . 17 (𝜑 → (∃𝑓𝐿𝑔𝐿 (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
4223, 41syl5bir 234 . . . . . . . . . . . . . . . 16 (𝜑 → ((∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
4322, 42impbid2 217 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣)))
44 df-ima 5324 . . . . . . . . . . . . . . . . . . 19 (𝐻) = ran (𝐻)
45 filelss 21865 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝐿) → 𝑍)
4624, 45sylan 571 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐿) → 𝑍)
47 txflf.h . . . . . . . . . . . . . . . . . . . . . . 23 𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
4847reseq1i 5593 . . . . . . . . . . . . . . . . . . . . . 22 (𝐻) = ((𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ↾ )
49 resmpt 5654 . . . . . . . . . . . . . . . . . . . . . 22 (𝑍 → ((𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ↾ ) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5048, 49syl5eq 2852 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 → (𝐻) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5146, 50syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐿) → (𝐻) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5251rneqd 5554 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝐿) → ran (𝐻) = ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5344, 52syl5eq 2852 . . . . . . . . . . . . . . . . . 18 ((𝜑𝐿) → (𝐻) = ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5453sseq1d 3829 . . . . . . . . . . . . . . . . 17 ((𝜑𝐿) → ((𝐻) ⊆ (𝑢 × 𝑣) ↔ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣)))
55 opelxp 5346 . . . . . . . . . . . . . . . . . . 19 (⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ((𝐹𝑛) ∈ 𝑢 ∧ (𝐺𝑛) ∈ 𝑣))
5655ralbii 3168 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ∀𝑛 ((𝐹𝑛) ∈ 𝑢 ∧ (𝐺𝑛) ∈ 𝑣))
57 eqid 2806 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
5857fmpt 6598 . . . . . . . . . . . . . . . . . . 19 (∀𝑛 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩):⟶(𝑢 × 𝑣))
59 opex 5122 . . . . . . . . . . . . . . . . . . . . 21 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ V
6059, 57fnmpti 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) Fn
61 df-f 6101 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩):⟶(𝑢 × 𝑣) ↔ ((𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) Fn ∧ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣)))
6260, 61mpbiran 691 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩):⟶(𝑢 × 𝑣) ↔ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣))
6358, 62bitri 266 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣))
64 r19.26 3252 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ((𝐹𝑛) ∈ 𝑢 ∧ (𝐺𝑛) ∈ 𝑣) ↔ (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
6556, 63, 643bitr3i 292 . . . . . . . . . . . . . . . . 17 (ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣) ↔ (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
6654, 65syl6bb 278 . . . . . . . . . . . . . . . 16 ((𝜑𝐿) → ((𝐻) ⊆ (𝑢 × 𝑣) ↔ (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
6766rexbidva 3237 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣) ↔ ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
68 txflf.f . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹:𝑍𝑋)
6968adantr 468 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝐿) → 𝐹:𝑍𝑋)
7069ffund 6256 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓𝐿) → Fun 𝐹)
71 filelss 21865 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑓𝐿) → 𝑓𝑍)
7224, 71sylan 571 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝐿) → 𝑓𝑍)
7369fdmd 6261 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝐿) → dom 𝐹 = 𝑍)
7472, 73sseqtr4d 3839 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓𝐿) → 𝑓 ⊆ dom 𝐹)
75 funimass4 6464 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐹𝑓 ⊆ dom 𝐹) → ((𝐹𝑓) ⊆ 𝑢 ↔ ∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
7670, 74, 75syl2anc 575 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝐿) → ((𝐹𝑓) ⊆ 𝑢 ↔ ∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
7776rexbidva 3237 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ↔ ∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
78 txflf.g . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺:𝑍𝑌)
7978adantr 468 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑔𝐿) → 𝐺:𝑍𝑌)
8079ffund 6256 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑔𝐿) → Fun 𝐺)
81 filelss 21865 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑔𝐿) → 𝑔𝑍)
8224, 81sylan 571 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑔𝐿) → 𝑔𝑍)
8379fdmd 6261 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑔𝐿) → dom 𝐺 = 𝑍)
8482, 83sseqtr4d 3839 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑔𝐿) → 𝑔 ⊆ dom 𝐺)
85 funimass4 6464 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐺𝑔 ⊆ dom 𝐺) → ((𝐺𝑔) ⊆ 𝑣 ↔ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
8680, 84, 85syl2anc 575 . . . . . . . . . . . . . . . . 17 ((𝜑𝑔𝐿) → ((𝐺𝑔) ⊆ 𝑣 ↔ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
8786rexbidva 3237 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣 ↔ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
8877, 87anbi12d 618 . . . . . . . . . . . . . . 15 (𝜑 → ((∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣)))
8943, 67, 883bitr4d 302 . . . . . . . . . . . . . 14 (𝜑 → (∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣) ↔ (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
9015, 89imbi12d 335 . . . . . . . . . . . . 13 (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ((𝑆𝑣𝑅𝑢) → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
91 impexp 439 . . . . . . . . . . . . 13 (((𝑆𝑣𝑅𝑢) → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)) ↔ (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
9290, 91syl6bb 278 . . . . . . . . . . . 12 (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
9392ralbidv 3174 . . . . . . . . . . 11 (𝜑 → (∀𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑣𝐾 (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
94 eleq2 2874 . . . . . . . . . . . . 13 (𝑥 = 𝑣 → (𝑆𝑥𝑆𝑣))
9594ralrab 3564 . . . . . . . . . . . 12 (∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)) ↔ ∀𝑣𝐾 (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
96 r19.21v 3148 . . . . . . . . . . . 12 (∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)) ↔ (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
9795, 96bitr3i 268 . . . . . . . . . . 11 (∀𝑣𝐾 (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))) ↔ (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
9893, 97syl6bb 278 . . . . . . . . . 10 (𝜑 → (∀𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
9998ralbidv 3174 . . . . . . . . 9 (𝜑 → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢𝐽 (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
100 eleq2 2874 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑅𝑥𝑅𝑢))
101100ralrab 3564 . . . . . . . . 9 (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ ∀𝑢𝐽 (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
10299, 101syl6bbr 280 . . . . . . . 8 (𝜑 → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
103102adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
104 txflf.j . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
105 toponmax 20940 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
106104, 105syl 17 . . . . . . . . . 10 (𝜑𝑋𝐽)
107 eleq2 2874 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑅𝑥𝑅𝑋))
108107rspcev 3502 . . . . . . . . . . 11 ((𝑋𝐽𝑅𝑋) → ∃𝑥𝐽 𝑅𝑥)
109 rabn0 4158 . . . . . . . . . . 11 ({𝑥𝐽𝑅𝑥} ≠ ∅ ↔ ∃𝑥𝐽 𝑅𝑥)
110108, 109sylibr 225 . . . . . . . . . 10 ((𝑋𝐽𝑅𝑋) → {𝑥𝐽𝑅𝑥} ≠ ∅)
111106, 110sylan 571 . . . . . . . . 9 ((𝜑𝑅𝑋) → {𝑥𝐽𝑅𝑥} ≠ ∅)
112 txflf.k . . . . . . . . . . 11 (𝜑𝐾 ∈ (TopOn‘𝑌))
113 toponmax 20940 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
114112, 113syl 17 . . . . . . . . . 10 (𝜑𝑌𝐾)
115 eleq2 2874 . . . . . . . . . . . 12 (𝑥 = 𝑌 → (𝑆𝑥𝑆𝑌))
116115rspcev 3502 . . . . . . . . . . 11 ((𝑌𝐾𝑆𝑌) → ∃𝑥𝐾 𝑆𝑥)
117 rabn0 4158 . . . . . . . . . . 11 ({𝑥𝐾𝑆𝑥} ≠ ∅ ↔ ∃𝑥𝐾 𝑆𝑥)
118116, 117sylibr 225 . . . . . . . . . 10 ((𝑌𝐾𝑆𝑌) → {𝑥𝐾𝑆𝑥} ≠ ∅)
119114, 118sylan 571 . . . . . . . . 9 ((𝜑𝑆𝑌) → {𝑥𝐾𝑆𝑥} ≠ ∅)
120111, 119anim12dan 607 . . . . . . . 8 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → ({𝑥𝐽𝑅𝑥} ≠ ∅ ∧ {𝑥𝐾𝑆𝑥} ≠ ∅))
121 r19.28zv 4261 . . . . . . . . . 10 ({𝑥𝐾𝑆𝑥} ≠ ∅ → (∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
122121ralbidv 3174 . . . . . . . . 9 ({𝑥𝐾𝑆𝑥} ≠ ∅ → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ ∀𝑢 ∈ {𝑥𝐽𝑅𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
123 r19.27zv 4266 . . . . . . . . 9 ({𝑥𝐽𝑅𝑥} ≠ ∅ → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
124122, 123sylan9bbr 502 . . . . . . . 8 (({𝑥𝐽𝑅𝑥} ≠ ∅ ∧ {𝑥𝐾𝑆𝑥} ≠ ∅) → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
125120, 124syl 17 . . . . . . 7 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
126103, 125bitrd 270 . . . . . 6 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
127100ralrab 3564 . . . . . . 7 (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ↔ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢))
12894ralrab 3564 . . . . . . 7 (∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣 ↔ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))
129127, 128anbi12i 614 . . . . . 6 ((∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
130126, 129syl6bb 278 . . . . 5 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
13111, 130syl5bb 274 . . . 4 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧) ↔ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
132131pm5.32da 570 . . 3 (𝜑 → (((𝑅𝑋𝑆𝑌) ∧ ∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧)) ↔ ((𝑅𝑋𝑆𝑌) ∧ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
133 opelxp 5346 . . . 4 (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ↔ (𝑅𝑋𝑆𝑌))
134133anbi1i 612 . . 3 ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧)) ↔ ((𝑅𝑋𝑆𝑌) ∧ ∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧)))
135 an4 638 . . 3 (((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))) ↔ ((𝑅𝑋𝑆𝑌) ∧ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
136132, 134, 1353bitr4g 305 . 2 (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧)) ↔ ((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
137 eqid 2806 . . . . . . . 8 ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
138137txval 21577 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
139104, 112, 138syl2anc 575 . . . . . 6 (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
140139oveq1d 6885 . . . . 5 (𝜑 → ((𝐽 ×t 𝐾) fLimf 𝐿) = ((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿))
141140fveq1d 6406 . . . 4 (𝜑 → (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) = (((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻))
142141eleq2d 2871 . . 3 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ ⟨𝑅, 𝑆⟩ ∈ (((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻)))
143 txtopon 21604 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
144104, 112, 143syl2anc 575 . . . . 5 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
145139, 144eqeltrrd 2886 . . . 4 (𝜑 → (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) ∈ (TopOn‘(𝑋 × 𝑌)))
14668ffvelrnda 6577 . . . . . 6 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ 𝑋)
14778ffvelrnda 6577 . . . . . 6 ((𝜑𝑛𝑍) → (𝐺𝑛) ∈ 𝑌)
148 opelxpi 5348 . . . . . 6 (((𝐹𝑛) ∈ 𝑋 ∧ (𝐺𝑛) ∈ 𝑌) → ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑋 × 𝑌))
149146, 147, 148syl2anc 575 . . . . 5 ((𝜑𝑛𝑍) → ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑋 × 𝑌))
150149, 47fmptd 6602 . . . 4 (𝜑𝐻:𝑍⟶(𝑋 × 𝑌))
151 eqid 2806 . . . . 5 (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)))
152151flftg 22009 . . . 4 (((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐻:𝑍⟶(𝑋 × 𝑌)) → (⟨𝑅, 𝑆⟩ ∈ (((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧))))
153145, 24, 150, 152syl3anc 1483 . . 3 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧))))
154142, 153bitrd 270 . 2 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧))))
155 isflf 22006 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐹:𝑍𝑋) → (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢))))
156104, 24, 68, 155syl3anc 1483 . . 3 (𝜑 → (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢))))
157 isflf 22006 . . . 4 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺) ↔ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
158112, 24, 78, 157syl3anc 1483 . . 3 (𝜑 → (𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺) ↔ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
159156, 158anbi12d 618 . 2 (𝜑 → ((𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺)) ↔ ((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
160136, 154, 1593bitr4d 302 1 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  wne 2978  wral 3096  wrex 3097  {crab 3100  Vcvv 3391  cin 3768  wss 3769  c0 4116  cop 4376  cmpt 4923   × cxp 5309  dom cdm 5311  ran crn 5312  cres 5313  cima 5314  Fun wfun 6091   Fn wfn 6092  wf 6093  cfv 6097  (class class class)co 6870  cmpt2 6872  topGenctg 16299  TopOnctopon 20924   ×t ctx 21573  Filcfil 21858   fLimf cflf 21948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-1st 7394  df-2nd 7395  df-map 8090  df-topgen 16305  df-fbas 19947  df-fg 19948  df-top 20908  df-topon 20925  df-bases 20960  df-ntr 21034  df-nei 21112  df-tx 21575  df-fil 21859  df-fm 21951  df-flim 21952  df-flf 21953
This theorem is referenced by:  flfcnp2  22020
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