MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  txflf Structured version   Visualization version   GIF version

Theorem txflf 24030
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 3482 . . . . . . . 8 𝑢 ∈ V
2 vex 3482 . . . . . . . 8 𝑣 ∈ V
31, 2xpex 7772 . . . . . . 7 (𝑢 × 𝑣) ∈ V
43rgen2w 3064 . . . . . 6 𝑢𝐽𝑣𝐾 (𝑢 × 𝑣) ∈ V
5 eqid 2735 . . . . . . 7 (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
6 eleq2 2828 . . . . . . . 8 (𝑧 = (𝑢 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑧 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣)))
7 sseq2 4022 . . . . . . . . 9 (𝑧 = (𝑢 × 𝑣) → ((𝐻) ⊆ 𝑧 ↔ (𝐻) ⊆ (𝑢 × 𝑣)))
87rexbidv 3177 . . . . . . . 8 (𝑧 = (𝑢 × 𝑣) → (∃𝐿 (𝐻) ⊆ 𝑧 ↔ ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)))
96, 8imbi12d 344 . . . . . . 7 (𝑧 = (𝑢 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃𝐿 (𝐻) ⊆ 𝑧) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣))))
105, 9ralrnmpo 7572 . . . . . 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 5725 . . . . . . . . . . . . . . . 16 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑅𝑢𝑆𝑣))
1312biancomi 462 . . . . . . . . . . . . . . 15 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑆𝑣𝑅𝑢))
1413a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑆𝑣𝑅𝑢)))
15 r19.40 3117 . . . . . . . . . . . . . . . . 17 (∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) → (∃𝐿𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝐿𝑛 (𝐺𝑛) ∈ 𝑣))
16 raleq 3321 . . . . . . . . . . . . . . . . . . 19 ( = 𝑓 → (∀𝑛 (𝐹𝑛) ∈ 𝑢 ↔ ∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
1716cbvrexvw 3236 . . . . . . . . . . . . . . . . . 18 (∃𝐿𝑛 (𝐹𝑛) ∈ 𝑢 ↔ ∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢)
18 raleq 3321 . . . . . . . . . . . . . . . . . . 19 ( = 𝑔 → (∀𝑛 (𝐺𝑛) ∈ 𝑣 ↔ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
1918cbvrexvw 3236 . . . . . . . . . . . . . . . . . 18 (∃𝐿𝑛 (𝐺𝑛) ∈ 𝑣 ↔ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣)
2017, 19anbi12i 628 . . . . . . . . . . . . . . . . 17 ((∃𝐿𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝐿𝑛 (𝐺𝑛) ∈ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
2115, 20sylib 218 . . . . . . . . . . . . . . . 16 (∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) → (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
22 reeanv 3227 . . . . . . . . . . . . . . . . 17 (∃𝑓𝐿𝑔𝐿 (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
23 txflf.l . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐿 ∈ (Fil‘𝑍))
24 filin 23878 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑓𝐿𝑔𝐿) → (𝑓𝑔) ∈ 𝐿)
25243expb 1119 . . . . . . . . . . . . . . . . . . . . 21 ((𝐿 ∈ (Fil‘𝑍) ∧ (𝑓𝐿𝑔𝐿)) → (𝑓𝑔) ∈ 𝐿)
2623, 25sylan 580 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓𝐿𝑔𝐿)) → (𝑓𝑔) ∈ 𝐿)
27 inss1 4245 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑔) ⊆ 𝑓
28 ssralv 4064 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑔) ⊆ 𝑓 → (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 → ∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢))
2927, 28ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 → ∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢)
30 inss2 4246 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑔) ⊆ 𝑔
31 ssralv 4064 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑔) ⊆ 𝑔 → (∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣 → ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣))
3230, 31ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣 → ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣)
3329, 32anim12i 613 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → (∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣))
34 raleq 3321 . . . . . . . . . . . . . . . . . . . . . 22 ( = (𝑓𝑔) → (∀𝑛 (𝐹𝑛) ∈ 𝑢 ↔ ∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢))
35 raleq 3321 . . . . . . . . . . . . . . . . . . . . . 22 ( = (𝑓𝑔) → (∀𝑛 (𝐺𝑛) ∈ 𝑣 ↔ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣))
3634, 35anbi12d 632 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑓𝑔) → ((∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) ↔ (∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣)))
3736rspcev 3622 . . . . . . . . . . . . . . . . . . . 20 (((𝑓𝑔) ∈ 𝐿 ∧ (∀𝑛 ∈ (𝑓𝑔)(𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓𝑔)(𝐺𝑛) ∈ 𝑣)) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
3826, 33, 37syl2an 596 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓𝐿𝑔𝐿)) ∧ (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣)) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
3938ex 412 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑓𝐿𝑔𝐿)) → ((∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
4039rexlimdvva 3211 . . . . . . . . . . . . . . . . 17 (𝜑 → (∃𝑓𝐿𝑔𝐿 (∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
4122, 40biimtrrid 243 . . . . . . . . . . . . . . . 16 (𝜑 → ((∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣) → ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
4221, 41impbid2 226 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣)))
43 df-ima 5702 . . . . . . . . . . . . . . . . . . 19 (𝐻) = ran (𝐻)
44 filelss 23876 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝐿) → 𝑍)
4523, 44sylan 580 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐿) → 𝑍)
46 txflf.h . . . . . . . . . . . . . . . . . . . . . . 23 𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
4746reseq1i 5996 . . . . . . . . . . . . . . . . . . . . . 22 (𝐻) = ((𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ↾ )
48 resmpt 6057 . . . . . . . . . . . . . . . . . . . . . 22 (𝑍 → ((𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ↾ ) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
4947, 48eqtrid 2787 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 → (𝐻) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5045, 49syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐿) → (𝐻) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5150rneqd 5952 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝐿) → ran (𝐻) = ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5243, 51eqtrid 2787 . . . . . . . . . . . . . . . . . 18 ((𝜑𝐿) → (𝐻) = ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩))
5352sseq1d 4027 . . . . . . . . . . . . . . . . 17 ((𝜑𝐿) → ((𝐻) ⊆ (𝑢 × 𝑣) ↔ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣)))
54 opelxp 5725 . . . . . . . . . . . . . . . . . . 19 (⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ((𝐹𝑛) ∈ 𝑢 ∧ (𝐺𝑛) ∈ 𝑣))
5554ralbii 3091 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ∀𝑛 ((𝐹𝑛) ∈ 𝑢 ∧ (𝐺𝑛) ∈ 𝑣))
56 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) = (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
5756fmpt 7130 . . . . . . . . . . . . . . . . . . 19 (∀𝑛 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩):⟶(𝑢 × 𝑣))
58 opex 5475 . . . . . . . . . . . . . . . . . . . . 21 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ V
5958, 56fnmpti 6712 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) Fn
60 df-f 6567 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩):⟶(𝑢 × 𝑣) ↔ ((𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) Fn ∧ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣)))
6159, 60mpbiran 709 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩):⟶(𝑢 × 𝑣) ↔ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣))
6257, 61bitri 275 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣))
63 r19.26 3109 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ((𝐹𝑛) ∈ 𝑢 ∧ (𝐺𝑛) ∈ 𝑣) ↔ (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
6455, 62, 633bitr3i 301 . . . . . . . . . . . . . . . . 17 (ran (𝑛 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩) ⊆ (𝑢 × 𝑣) ↔ (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣))
6553, 64bitrdi 287 . . . . . . . . . . . . . . . 16 ((𝜑𝐿) → ((𝐻) ⊆ (𝑢 × 𝑣) ↔ (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
6665rexbidva 3175 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣) ↔ ∃𝐿 (∀𝑛 (𝐹𝑛) ∈ 𝑢 ∧ ∀𝑛 (𝐺𝑛) ∈ 𝑣)))
67 txflf.f . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹:𝑍𝑋)
6867adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝐿) → 𝐹:𝑍𝑋)
6968ffund 6741 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓𝐿) → Fun 𝐹)
70 filelss 23876 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑓𝐿) → 𝑓𝑍)
7123, 70sylan 580 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝐿) → 𝑓𝑍)
7268fdmd 6747 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝐿) → dom 𝐹 = 𝑍)
7371, 72sseqtrrd 4037 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓𝐿) → 𝑓 ⊆ dom 𝐹)
74 funimass4 6973 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐹𝑓 ⊆ dom 𝐹) → ((𝐹𝑓) ⊆ 𝑢 ↔ ∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
7569, 73, 74syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝐿) → ((𝐹𝑓) ⊆ 𝑢 ↔ ∀𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
7675rexbidva 3175 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ↔ ∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢))
77 txflf.g . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺:𝑍𝑌)
7877adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑔𝐿) → 𝐺:𝑍𝑌)
7978ffund 6741 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑔𝐿) → Fun 𝐺)
80 filelss 23876 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑔𝐿) → 𝑔𝑍)
8123, 80sylan 580 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑔𝐿) → 𝑔𝑍)
8278fdmd 6747 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑔𝐿) → dom 𝐺 = 𝑍)
8381, 82sseqtrrd 4037 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑔𝐿) → 𝑔 ⊆ dom 𝐺)
84 funimass4 6973 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐺𝑔 ⊆ dom 𝐺) → ((𝐺𝑔) ⊆ 𝑣 ↔ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
8579, 83, 84syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑔𝐿) → ((𝐺𝑔) ⊆ 𝑣 ↔ ∀𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
8685rexbidva 3175 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣 ↔ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣))
8776, 86anbi12d 632 . . . . . . . . . . . . . . 15 (𝜑 → ((∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∃𝑓𝐿𝑛𝑓 (𝐹𝑛) ∈ 𝑢 ∧ ∃𝑔𝐿𝑛𝑔 (𝐺𝑛) ∈ 𝑣)))
8842, 66, 873bitr4d 311 . . . . . . . . . . . . . 14 (𝜑 → (∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣) ↔ (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
8914, 88imbi12d 344 . . . . . . . . . . . . 13 (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ((𝑆𝑣𝑅𝑢) → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
90 impexp 450 . . . . . . . . . . . . 13 (((𝑆𝑣𝑅𝑢) → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)) ↔ (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
9189, 90bitrdi 287 . . . . . . . . . . . 12 (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
9291ralbidv 3176 . . . . . . . . . . 11 (𝜑 → (∀𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑣𝐾 (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
93 eleq2 2828 . . . . . . . . . . . . 13 (𝑥 = 𝑣 → (𝑆𝑥𝑆𝑣))
9493ralrab 3702 . . . . . . . . . . . 12 (∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)) ↔ ∀𝑣𝐾 (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
95 r19.21v 3178 . . . . . . . . . . . 12 (∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)) ↔ (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
9694, 95bitr3i 277 . . . . . . . . . . 11 (∀𝑣𝐾 (𝑆𝑣 → (𝑅𝑢 → (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))) ↔ (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
9792, 96bitrdi 287 . . . . . . . . . 10 (𝜑 → (∀𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
9897ralbidv 3176 . . . . . . . . 9 (𝜑 → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢𝐽 (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
99 eleq2 2828 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑅𝑥𝑅𝑢))
10099ralrab 3702 . . . . . . . . 9 (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ ∀𝑢𝐽 (𝑅𝑢 → ∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
10198, 100bitr4di 289 . . . . . . . 8 (𝜑 → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
102101adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
103 txflf.j . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
104 toponmax 22948 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
105103, 104syl 17 . . . . . . . . . 10 (𝜑𝑋𝐽)
106 eleq2 2828 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑅𝑥𝑅𝑋))
107106rspcev 3622 . . . . . . . . . . 11 ((𝑋𝐽𝑅𝑋) → ∃𝑥𝐽 𝑅𝑥)
108 rabn0 4395 . . . . . . . . . . 11 ({𝑥𝐽𝑅𝑥} ≠ ∅ ↔ ∃𝑥𝐽 𝑅𝑥)
109107, 108sylibr 234 . . . . . . . . . 10 ((𝑋𝐽𝑅𝑋) → {𝑥𝐽𝑅𝑥} ≠ ∅)
110105, 109sylan 580 . . . . . . . . 9 ((𝜑𝑅𝑋) → {𝑥𝐽𝑅𝑥} ≠ ∅)
111 txflf.k . . . . . . . . . . 11 (𝜑𝐾 ∈ (TopOn‘𝑌))
112 toponmax 22948 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
113111, 112syl 17 . . . . . . . . . 10 (𝜑𝑌𝐾)
114 eleq2 2828 . . . . . . . . . . . 12 (𝑥 = 𝑌 → (𝑆𝑥𝑆𝑌))
115114rspcev 3622 . . . . . . . . . . 11 ((𝑌𝐾𝑆𝑌) → ∃𝑥𝐾 𝑆𝑥)
116 rabn0 4395 . . . . . . . . . . 11 ({𝑥𝐾𝑆𝑥} ≠ ∅ ↔ ∃𝑥𝐾 𝑆𝑥)
117115, 116sylibr 234 . . . . . . . . . 10 ((𝑌𝐾𝑆𝑌) → {𝑥𝐾𝑆𝑥} ≠ ∅)
118113, 117sylan 580 . . . . . . . . 9 ((𝜑𝑆𝑌) → {𝑥𝐾𝑆𝑥} ≠ ∅)
119110, 118anim12dan 619 . . . . . . . 8 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → ({𝑥𝐽𝑅𝑥} ≠ ∅ ∧ {𝑥𝐾𝑆𝑥} ≠ ∅))
120 r19.28zv 4507 . . . . . . . . . 10 ({𝑥𝐾𝑆𝑥} ≠ ∅ → (∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
121120ralbidv 3176 . . . . . . . . 9 ({𝑥𝐾𝑆𝑥} ≠ ∅ → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ ∀𝑢 ∈ {𝑥𝐽𝑅𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
122 r19.27zv 4512 . . . . . . . . 9 ({𝑥𝐽𝑅𝑥} ≠ ∅ → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
123121, 122sylan9bbr 510 . . . . . . . 8 (({𝑥𝐽𝑅𝑥} ≠ ∅ ∧ {𝑥𝐾𝑆𝑥} ≠ ∅) → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
124119, 123syl 17 . . . . . . 7 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∀𝑣 ∈ {𝑥𝐾𝑆𝑥} (∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
125102, 124bitrd 279 . . . . . 6 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
12699ralrab 3702 . . . . . . 7 (∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ↔ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢))
12793ralrab 3702 . . . . . . 7 (∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣 ↔ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))
128126, 127anbi12i 628 . . . . . 6 ((∀𝑢 ∈ {𝑥𝐽𝑅𝑥}∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥𝐾𝑆𝑥}∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣) ↔ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))
129125, 128bitrdi 287 . . . . 5 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃𝐿 (𝐻) ⊆ (𝑢 × 𝑣)) ↔ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
13011, 129bitrid 283 . . . 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))))
131130pm5.32da 579 . . 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)))))
132 opelxp 5725 . . . 4 (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ↔ (𝑅𝑋𝑆𝑌))
133132anbi1i 624 . . 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)))
134 an4 656 . . 3 (((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))) ↔ ((𝑅𝑋𝑆𝑌) ∧ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
135131, 133, 1343bitr4g 314 . 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)))))
136 eqid 2735 . . . . . . . 8 ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
137136txval 23588 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
138103, 111, 137syl2anc 584 . . . . . 6 (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
139138oveq1d 7446 . . . . 5 (𝜑 → ((𝐽 ×t 𝐾) fLimf 𝐿) = ((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿))
140139fveq1d 6909 . . . 4 (𝜑 → (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) = (((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻))
141140eleq2d 2825 . . 3 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ ⟨𝑅, 𝑆⟩ ∈ (((topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻)))
142 txtopon 23615 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
143103, 111, 142syl2anc 584 . . . . 5 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
144138, 143eqeltrrd 2840 . . . 4 (𝜑 → (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) ∈ (TopOn‘(𝑋 × 𝑌)))
14567ffvelcdmda 7104 . . . . . 6 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ 𝑋)
14677ffvelcdmda 7104 . . . . . 6 ((𝜑𝑛𝑍) → (𝐺𝑛) ∈ 𝑌)
147145, 146opelxpd 5728 . . . . 5 ((𝜑𝑛𝑍) → ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑋 × 𝑌))
148147, 46fmptd 7134 . . . 4 (𝜑𝐻:𝑍⟶(𝑋 × 𝑌))
149 eqid 2735 . . . . 5 (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)))
150149flftg 24020 . . . 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))))
151144, 23, 148, 150syl3anc 1370 . . 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))))
152141, 151bitrd 279 . 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))))
153 isflf 24017 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐹:𝑍𝑋) → (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢))))
154103, 23, 67, 153syl3anc 1370 . . 3 (𝜑 → (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢))))
155 isflf 24017 . . . 4 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺) ↔ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
156111, 23, 77, 155syl3anc 1370 . . 3 (𝜑 → (𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺) ↔ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣))))
157154, 156anbi12d 632 . 2 (𝜑 → ((𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺)) ↔ ((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑓𝐿 (𝐹𝑓) ⊆ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑔𝐿 (𝐺𝑔) ⊆ 𝑣)))))
158135, 152, 1573bitr4d 311 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 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cin 3962  wss 3963  c0 4339  cop 4637  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  topGenctg 17484  TopOnctopon 22932   ×t ctx 23584  Filcfil 23869   fLimf cflf 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-bases 22969  df-ntr 23044  df-nei 23122  df-tx 23586  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964
This theorem is referenced by:  flfcnp2  24031
  Copyright terms: Public domain W3C validator