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Theorem txflf 21857
 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 3234 . . . . . . . 8 𝑢 ∈ V
2 vex 3234 . . . . . . . 8 𝑣 ∈ V
31, 2xpex 7004 . . . . . . 7 (𝑢 × 𝑣) ∈ V
43rgen2w 2954 . . . . . 6 𝑢𝐽𝑣𝐾 (𝑢 × 𝑣) ∈ V
5 eqid 2651 . . . . . . 7 (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
6 eleq2 2719 . . . . . . . 8 (𝑧 = (𝑢 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑧 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣)))
7 sseq2 3660 . . . . . . . . 9 (𝑧 = (𝑢 × 𝑣) → ((𝐻) ⊆ 𝑧 ↔ (𝐻) ⊆ (𝑢 × 𝑣)))
87rexbidv 3081 . . . . . . . 8 (𝑧 = (𝑢 × 𝑣) → (∃𝐿 (𝐻) ⊆ 𝑧 ↔ ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)))
96, 8imbi12d 333 . . . . . . 7 (𝑧 = (𝑢 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣))))
105, 9ralrnmpt2 6817 . . . . . 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 5180 . . . . . . . . . . . . . . . 16 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑅𝑢𝑆𝑣))
13 ancom 465 . . . . . . . . . . . . . . . 16 ((𝑅𝑢𝑆𝑣) ↔ (𝑆𝑣𝑅𝑢))
1412, 13bitri 264 . . . . . . . . . . . . . . 15 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑆𝑣𝑅𝑢))
1514a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑆𝑣𝑅𝑢)))
16 r19.40 3117 . . . . . . . . . . . . . . . . 17 (∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) → (∃𝐿𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝐿𝑛 (𝐺𝑛) ∈ 𝑣))
17 raleq 3168 . . . . . . . . . . . . . . . . . . 19 ( = 𝑓 → (∀𝑛 (𝐹𝑛) ∈ 𝑢 ↔ ∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
1817cbvrexv 3202 . . . . . . . . . . . . . . . . . 18 (∃𝐿𝑛 (𝐹𝑛) ∈ 𝑢 ↔ ∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢)
19 raleq 3168 . . . . . . . . . . . . . . . . . . 19 ( = 𝑔 → (∀𝑛 (𝐺𝑛) ∈ 𝑣 ↔ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
2019cbvrexv 3202 . . . . . . . . . . . . . . . . . 18 (∃𝐿𝑛 (𝐺𝑛) ∈ 𝑣 ↔ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣)
2118, 20anbi12i 733 . . . . . . . . . . . . . . . . 17 ((∃𝐿𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝐿𝑛 (𝐺𝑛) ∈ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
2216, 21sylib 208 . . . . . . . . . . . . . . . 16 (∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) → (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
23 reeanv 3136 . . . . . . . . . . . . . . . . 17 (∃𝑓𝐿𝑔𝐿 (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
24 txflf.l . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐿 ∈ (Fil‘𝑍))
25 filin 21705 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑓𝐿𝑔𝐿) → (𝑓𝑔) ∈ 𝐿)
26253expb 1285 . . . . . . . . . . . . . . . . . . . . 21 ((𝐿 ∈ (Fil‘𝑍) ∧ (𝑓𝐿𝑔𝐿)) → (𝑓𝑔) ∈ 𝐿)
2724, 26sylan 487 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓𝐿𝑔𝐿)) → (𝑓𝑔) ∈ 𝐿)
28 inss1 3866 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑔) ⊆ 𝑓
29 ssralv 3699 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑔) ⊆ 𝑓 → (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 → ∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢))
3028, 29ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 → ∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢)
31 inss2 3867 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑔) ⊆ 𝑔
32 ssralv 3699 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑔) ⊆ 𝑔 → (∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣 → ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣))
3331, 32ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣 → ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣)
3430, 33anim12i 589 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → (∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣))
35 raleq 3168 . . . . . . . . . . . . . . . . . . . . . 22 ( = (𝑓𝑔) → (∀𝑛 (𝐹𝑛) ∈ 𝑢 ↔ ∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢))
36 raleq 3168 . . . . . . . . . . . . . . . . . . . . . 22 ( = (𝑓𝑔) → (∀𝑛 (𝐺𝑛) ∈ 𝑣 ↔ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣))
3735, 36anbi12d 747 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑓𝑔) → ((∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) ↔ (∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣)))
3837rspcev 3340 . . . . . . . . . . . . . . . . . . . 20 (((𝑓𝑔) ∈ 𝐿 ∧ (∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣)) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
3927, 34, 38syl2an 493 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓𝐿𝑔𝐿)) ∧ (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣)) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
4039ex 449 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑓𝐿𝑔𝐿)) → ((∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
4140rexlimdvva 3067 . . . . . . . . . . . . . . . . 17 (𝜑 → (∃𝑓𝐿𝑔𝐿 (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
4223, 41syl5bir 233 . . . . . . . . . . . . . . . 16 (𝜑 → ((∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
4322, 42impbid2 216 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣)))
44 df-ima 5156 . . . . . . . . . . . . . . . . . . 19 (𝐻) = ran (𝐻)
45 filelss 21703 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝐿) → 𝑍)
4624, 45sylan 487 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐿) → 𝑍)
47 txflf.h . . . . . . . . . . . . . . . . . . . . . . 23 𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
4847reseq1i 5424 . . . . . . . . . . . . . . . . . . . . . 22 (𝐻) = ((𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ↾ )
49 resmpt 5484 . . . . . . . . . . . . . . . . . . . . . 22 (𝑍 → ((𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ↾ ) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5048, 49syl5eq 2697 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 → (𝐻) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5146, 50syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐿) → (𝐻) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5251rneqd 5385 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝐿) → ran (𝐻) = ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5344, 52syl5eq 2697 . . . . . . . . . . . . . . . . . 18 ((𝜑𝐿) → (𝐻) = ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5453sseq1d 3665 . . . . . . . . . . . . . . . . 17 ((𝜑𝐿) → ((𝐻) ⊆ (𝑢 × 𝑣) ↔ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣)))
55 opelxp 5180 . . . . . . . . . . . . . . . . . . 19 (⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ((𝐹𝑛) ∈ 𝑢 ∧ (𝐺𝑛) ∈ 𝑣))
5655ralbii 3009 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ∀𝑛 ((𝐹𝑛) ∈ 𝑢 ∧ (𝐺𝑛) ∈ 𝑣))
57 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
5857fmpt 6421 . . . . . . . . . . . . . . . . . . 19 (∀𝑛 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩):⟶(𝑢 × 𝑣))
59 opex 4962 . . . . . . . . . . . . . . . . . . . . 21 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ V
6059, 57fnmpti 6060 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) Fn
61 df-f 5930 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩):⟶(𝑢 × 𝑣) ↔ ((𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) Fn ∧ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣)))
6260, 61mpbiran 973 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩):⟶(𝑢 × 𝑣) ↔ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣))
6358, 62bitri 264 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣))
64 r19.26 3093 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ((𝐹𝑛) ∈ 𝑢 ∧ (𝐺𝑛) ∈ 𝑣) ↔ (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
6556, 63, 643bitr3i 290 . . . . . . . . . . . . . . . . 17 (ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣) ↔ (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
6654, 65syl6bb 276 . . . . . . . . . . . . . . . 16 ((𝜑𝐿) → ((𝐻) ⊆ (𝑢 × 𝑣) ↔ (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
6766rexbidva 3078 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣) ↔ ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
68 txflf.f . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹:𝑍𝑋)
6968adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝐿) → 𝐹:𝑍𝑋)
70 ffun 6086 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝑍𝑋 → Fun 𝐹)
7169, 70syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓𝐿) → Fun 𝐹)
72 filelss 21703 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑓𝐿) → 𝑓𝑍)
7324, 72sylan 487 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝐿) → 𝑓𝑍)
74 fdm 6089 . . . . . . . . . . . . . . . . . . . 20 (𝐹:𝑍𝑋 → dom 𝐹 = 𝑍)
7569, 74syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝐿) → dom 𝐹 = 𝑍)
7673, 75sseqtr4d 3675 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓𝐿) → 𝑓 ⊆ dom 𝐹)
77 funimass4 6286 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐹𝑓 ⊆ dom 𝐹) → ((𝐹𝑓) ⊆ 𝑢 ↔ ∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
7871, 76, 77syl2anc 694 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝐿) → ((𝐹𝑓) ⊆ 𝑢 ↔ ∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
7978rexbidva 3078 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ↔ ∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
80 txflf.g . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺:𝑍𝑌)
8180adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑔𝐿) → 𝐺:𝑍𝑌)
82 ffun 6086 . . . . . . . . . . . . . . . . . . 19 (𝐺:𝑍𝑌 → Fun 𝐺)
8381, 82syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑔𝐿) → Fun 𝐺)
84 filelss 21703 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑔𝐿) → 𝑔𝑍)
8524, 84sylan 487 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑔𝐿) → 𝑔𝑍)
86 fdm 6089 . . . . . . . . . . . . . . . . . . . 20 (𝐺:𝑍𝑌 → dom 𝐺 = 𝑍)
8781, 86syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑔𝐿) → dom 𝐺 = 𝑍)
8885, 87sseqtr4d 3675 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑔𝐿) → 𝑔 ⊆ dom 𝐺)
89 funimass4 6286 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐺𝑔 ⊆ dom 𝐺) → ((𝐺𝑔) ⊆ 𝑣 ↔ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
9083, 88, 89syl2anc 694 . . . . . . . . . . . . . . . . 17 ((𝜑𝑔𝐿) → ((𝐺𝑔) ⊆ 𝑣 ↔ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
9190rexbidva 3078 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣 ↔ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
9279, 91anbi12d 747 . . . . . . . . . . . . . . 15 (𝜑 → ((∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣)))
9343, 67, 923bitr4d 300 . . . . . . . . . . . . . 14 (𝜑 → (∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣) ↔ (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
9415, 93imbi12d 333 . . . . . . . . . . . . 13 (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ((𝑆𝑣𝑅𝑢) → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
95 impexp 461 . . . . . . . . . . . . 13 (((𝑆𝑣𝑅𝑢) → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)) ↔ (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
9694, 95syl6bb 276 . . . . . . . . . . . 12 (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
9796ralbidv 3015 . . . . . . . . . . 11 (𝜑 → (∀𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑣𝐾 (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
98 eleq2 2719 . . . . . . . . . . . . 13 (𝑥 = 𝑣 → (𝑆𝑥𝑆𝑣))
9998ralrab 3401 . . . . . . . . . . . 12 (∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)) ↔ ∀𝑣𝐾 (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
100 r19.21v 2989 . . . . . . . . . . . 12 (∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)) ↔ (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
10199, 100bitr3i 266 . . . . . . . . . . 11 (∀𝑣𝐾 (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))) ↔ (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
10297, 101syl6bb 276 . . . . . . . . . 10 (𝜑 → (∀𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
103102ralbidv 3015 . . . . . . . . 9 (𝜑 → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢𝐽 (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
104 eleq2 2719 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑅𝑥𝑅𝑢))
105104ralrab 3401 . . . . . . . . 9 (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ ∀𝑢𝐽 (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
106103, 105syl6bbr 278 . . . . . . . 8 (𝜑 → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
107106adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
108 txflf.j . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
109 toponmax 20778 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
110108, 109syl 17 . . . . . . . . . 10 (𝜑𝑋𝐽)
111 eleq2 2719 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑅𝑥𝑅𝑋))
112111rspcev 3340 . . . . . . . . . . 11 ((𝑋𝐽𝑅𝑋) → ∃𝑥𝐽 𝑅𝑥)
113 rabn0 3991 . . . . . . . . . . 11 ({𝑥𝐽𝑅𝑥} ≠ ∅ ↔ ∃𝑥𝐽 𝑅𝑥)
114112, 113sylibr 224 . . . . . . . . . 10 ((𝑋𝐽𝑅𝑋) → {𝑥𝐽𝑅𝑥} ≠ ∅)
115110, 114sylan 487 . . . . . . . . 9 ((𝜑𝑅𝑋) → {𝑥𝐽𝑅𝑥} ≠ ∅)
116 txflf.k . . . . . . . . . . 11 (𝜑𝐾 ∈ (TopOn‘𝑌))
117 toponmax 20778 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
118116, 117syl 17 . . . . . . . . . 10 (𝜑𝑌𝐾)
119 eleq2 2719 . . . . . . . . . . . 12 (𝑥 = 𝑌 → (𝑆𝑥𝑆𝑌))
120119rspcev 3340 . . . . . . . . . . 11 ((𝑌𝐾𝑆𝑌) → ∃𝑥𝐾 𝑆𝑥)
121 rabn0 3991 . . . . . . . . . . 11 ({𝑥𝐾𝑆𝑥} ≠ ∅ ↔ ∃𝑥𝐾 𝑆𝑥)
122120, 121sylibr 224 . . . . . . . . . 10 ((𝑌𝐾𝑆𝑌) → {𝑥𝐾𝑆𝑥} ≠ ∅)
123118, 122sylan 487 . . . . . . . . 9 ((𝜑𝑆𝑌) → {𝑥𝐾𝑆𝑥} ≠ ∅)
124115, 123anim12dan 900 . . . . . . . 8 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → ({𝑥𝐽𝑅𝑥} ≠ ∅ ∧ {𝑥𝐾𝑆𝑥} ≠ ∅))
125 r19.28zv 4099 . . . . . . . . . 10 ({𝑥𝐾𝑆𝑥} ≠ ∅ → (∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
126125ralbidv 3015 . . . . . . . . 9 ({𝑥𝐾𝑆𝑥} ≠ ∅ → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ ∀𝑢 ∈ {𝑥𝐽𝑅𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
127 r19.27zv 4104 . . . . . . . . 9 ({𝑥𝐽𝑅𝑥} ≠ ∅ → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
128126, 127sylan9bbr 737 . . . . . . . 8 (({𝑥𝐽𝑅𝑥} ≠ ∅ ∧ {𝑥𝐾𝑆𝑥} ≠ ∅) → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
129124, 128syl 17 . . . . . . 7 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
130107, 129bitrd 268 . . . . . 6 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
131104ralrab 3401 . . . . . . 7 (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ↔ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢))
13298ralrab 3401 . . . . . . 7 (∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣 ↔ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))
133131, 132anbi12i 733 . . . . . 6 ((∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
134130, 133syl6bb 276 . . . . 5 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
13511, 134syl5bb 272 . . . 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))))
136135pm5.32da 674 . . 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)))))
137 opelxp 5180 . . . 4 (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ↔ (𝑅𝑋𝑆𝑌))
138137anbi1i 731 . . 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)))
139 an4 882 . . 3 (((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))) ↔ ((𝑅𝑋𝑆𝑌) ∧ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
140136, 138, 1393bitr4g 303 . 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)))))
141 eqid 2651 . . . . . . . 8 ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
142141txval 21415 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
143108, 116, 142syl2anc 694 . . . . . 6 (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
144143oveq1d 6705 . . . . 5 (𝜑 → ((𝐽 ×t 𝐾) fLimf 𝐿) = ((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿))
145144fveq1d 6231 . . . 4 (𝜑 → (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) = (((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻))
146145eleq2d 2716 . . 3 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ ⟨𝑅, 𝑆⟩ ∈ (((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻)))
147 txtopon 21442 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
148108, 116, 147syl2anc 694 . . . . 5 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
149143, 148eqeltrrd 2731 . . . 4 (𝜑 → (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) ∈ (TopOn‘(𝑋 × 𝑌)))
15068ffvelrnda 6399 . . . . . 6 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ 𝑋)
15180ffvelrnda 6399 . . . . . 6 ((𝜑𝑛𝑍) → (𝐺𝑛) ∈ 𝑌)
152 opelxpi 5182 . . . . . 6 (((𝐹𝑛) ∈ 𝑋 ∧ (𝐺𝑛) ∈ 𝑌) → ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑋 × 𝑌))
153150, 151, 152syl2anc 694 . . . . 5 ((𝜑𝑛𝑍) → ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑋 × 𝑌))
154153, 47fmptd 6425 . . . 4 (𝜑𝐻:𝑍⟶(𝑋 × 𝑌))
155 eqid 2651 . . . . 5 (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)))
156155flftg 21847 . . . 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))))
157149, 24, 154, 156syl3anc 1366 . . 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))))
158146, 157bitrd 268 . 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))))
159 isflf 21844 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐹:𝑍𝑋) → (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢))))
160108, 24, 68, 159syl3anc 1366 . . 3 (𝜑 → (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢))))
161 isflf 21844 . . . 4 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺) ↔ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
162116, 24, 80, 161syl3anc 1366 . . 3 (𝜑 → (𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺) ↔ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
163160, 162anbi12d 747 . 2 (𝜑 → ((𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺)) ↔ ((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
164140, 158, 1633bitr4d 300 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 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ⟨cop 4216   ↦ cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144   ↾ cres 5145   “ cima 5146  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  topGenctg 16145  TopOnctopon 20763   ×t ctx 21411  Filcfil 21696   fLimf cflf 21786 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-topgen 16151  df-fbas 19791  df-fg 19792  df-top 20747  df-topon 20764  df-bases 20798  df-ntr 20872  df-nei 20950  df-tx 21413  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791 This theorem is referenced by:  flfcnp2  21858
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