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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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