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Theorem txtube 21814
Description: The "tube lemma". If 𝑋 is compact and there is an open set 𝑈 containing the line 𝑋 × {𝐴}, then there is a "tube" 𝑋 × 𝑢 for some neighborhood 𝑢 of 𝐴 which is entirely contained within 𝑈. (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypotheses
Ref Expression
txtube.x 𝑋 = 𝑅
txtube.y 𝑌 = 𝑆
txtube.r (𝜑𝑅 ∈ Comp)
txtube.s (𝜑𝑆 ∈ Top)
txtube.w (𝜑𝑈 ∈ (𝑅 ×t 𝑆))
txtube.u (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)
txtube.a (𝜑𝐴𝑌)
Assertion
Ref Expression
txtube (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑆   𝑢,𝑌   𝜑,𝑢   𝑢,𝑈   𝑢,𝑋

Proof of Theorem txtube
Dummy variables 𝑡 𝑓 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 txtube.r . . 3 (𝜑𝑅 ∈ Comp)
2 eleq1 2894 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝐴⟩ → (𝑦 ∈ (𝑢 × 𝑣) ↔ ⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣)))
32anbi1d 623 . . . . . . 7 (𝑦 = ⟨𝑥, 𝐴⟩ → ((𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
432rexbidv 3267 . . . . . 6 (𝑦 = ⟨𝑥, 𝐴⟩ → (∃𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑢𝑅𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
5 txtube.w . . . . . . . 8 (𝜑𝑈 ∈ (𝑅 ×t 𝑆))
6 txtube.s . . . . . . . . 9 (𝜑𝑆 ∈ Top)
7 eltx 21742 . . . . . . . . 9 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Top) → (𝑈 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
81, 6, 7syl2anc 579 . . . . . . . 8 (𝜑 → (𝑈 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
95, 8mpbid 224 . . . . . . 7 (𝜑 → ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
109adantr 474 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
11 txtube.u . . . . . . . 8 (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)
1211adantr 474 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑋 × {𝐴}) ⊆ 𝑈)
13 id 22 . . . . . . . 8 (𝑥𝑋𝑥𝑋)
14 txtube.a . . . . . . . . 9 (𝜑𝐴𝑌)
15 snidg 4427 . . . . . . . . 9 (𝐴𝑌𝐴 ∈ {𝐴})
1614, 15syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
17 opelxpi 5379 . . . . . . . 8 ((𝑥𝑋𝐴 ∈ {𝐴}) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
1813, 16, 17syl2anr 590 . . . . . . 7 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
1912, 18sseldd 3828 . . . . . 6 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑈)
204, 10, 19rspcdva 3532 . . . . 5 ((𝜑𝑥𝑋) → ∃𝑢𝑅𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
21 opelxp 5378 . . . . . . . . . 10 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ↔ (𝑥𝑢𝐴𝑣))
2221anbi1i 617 . . . . . . . . 9 ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ((𝑥𝑢𝐴𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
23 anass 462 . . . . . . . . 9 (((𝑥𝑢𝐴𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2422, 23bitri 267 . . . . . . . 8 ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2524rexbii 3251 . . . . . . 7 (∃𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑣𝑆 (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
26 r19.42v 3302 . . . . . . 7 (∃𝑣𝑆 (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)) ↔ (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2725, 26bitri 267 . . . . . 6 (∃𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2827rexbii 3251 . . . . 5 (∃𝑢𝑅𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2920, 28sylib 210 . . . 4 ((𝜑𝑥𝑋) → ∃𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
3029ralrimiva 3175 . . 3 (𝜑 → ∀𝑥𝑋𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
31 txtube.x . . . 4 𝑋 = 𝑅
32 eleq2 2895 . . . . 5 (𝑣 = (𝑓𝑢) → (𝐴𝑣𝐴 ∈ (𝑓𝑢)))
33 xpeq2 5363 . . . . . 6 (𝑣 = (𝑓𝑢) → (𝑢 × 𝑣) = (𝑢 × (𝑓𝑢)))
3433sseq1d 3857 . . . . 5 (𝑣 = (𝑓𝑢) → ((𝑢 × 𝑣) ⊆ 𝑈 ↔ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))
3532, 34anbi12d 624 . . . 4 (𝑣 = (𝑓𝑢) → ((𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))
3631, 35cmpcovf 21565 . . 3 ((𝑅 ∈ Comp ∧ ∀𝑥𝑋𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈))) → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))))
371, 30, 36syl2anc 579 . 2 (𝜑 → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))))
38 rint0 4737 . . . . . . . . . 10 (ran 𝑓 = ∅ → (𝑌 ran 𝑓) = 𝑌)
3938adantl 475 . . . . . . . . 9 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 = ∅) → (𝑌 ran 𝑓) = 𝑌)
40 txtube.y . . . . . . . . . . . 12 𝑌 = 𝑆
4140topopn 21081 . . . . . . . . . . 11 (𝑆 ∈ Top → 𝑌𝑆)
426, 41syl 17 . . . . . . . . . 10 (𝜑𝑌𝑆)
4342ad3antrrr 721 . . . . . . . . 9 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 = ∅) → 𝑌𝑆)
4439, 43eqeltrd 2906 . . . . . . . 8 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 = ∅) → (𝑌 ran 𝑓) ∈ 𝑆)
456ad3antrrr 721 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → 𝑆 ∈ Top)
46 simprrl 799 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑓:𝑡𝑆)
4746frnd 6285 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ran 𝑓𝑆)
4847adantr 474 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓𝑆)
49 simpr 479 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 ≠ ∅)
50 inss2 4058 . . . . . . . . . . . . . . . 16 (𝒫 𝑅 ∩ Fin) ⊆ Fin
51 simplr 785 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑡 ∈ (𝒫 𝑅 ∩ Fin))
5250, 51sseldi 3825 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑡 ∈ Fin)
5346ffnd 6279 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑓 Fn 𝑡)
54 dffn4 6359 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑡𝑓:𝑡onto→ran 𝑓)
5553, 54sylib 210 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑓:𝑡onto→ran 𝑓)
56 fofi 8521 . . . . . . . . . . . . . . 15 ((𝑡 ∈ Fin ∧ 𝑓:𝑡onto→ran 𝑓) → ran 𝑓 ∈ Fin)
5752, 55, 56syl2anc 579 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ran 𝑓 ∈ Fin)
5857adantr 474 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 ∈ Fin)
59 fiinopn 21076 . . . . . . . . . . . . . 14 (𝑆 ∈ Top → ((ran 𝑓𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin) → ran 𝑓𝑆))
6059imp 397 . . . . . . . . . . . . 13 ((𝑆 ∈ Top ∧ (ran 𝑓𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin)) → ran 𝑓𝑆)
6145, 48, 49, 58, 60syl13anc 1495 . . . . . . . . . . . 12 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓𝑆)
62 elssuni 4689 . . . . . . . . . . . 12 ( ran 𝑓𝑆 ran 𝑓 𝑆)
6361, 62syl 17 . . . . . . . . . . 11 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 𝑆)
6463, 40syl6sseqr 3877 . . . . . . . . . 10 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓𝑌)
65 sseqin2 4044 . . . . . . . . . 10 ( ran 𝑓𝑌 ↔ (𝑌 ran 𝑓) = ran 𝑓)
6664, 65sylib 210 . . . . . . . . 9 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → (𝑌 ran 𝑓) = ran 𝑓)
6766, 61eqeltrd 2906 . . . . . . . 8 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → (𝑌 ran 𝑓) ∈ 𝑆)
6844, 67pm2.61dane 3086 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝑌 ran 𝑓) ∈ 𝑆)
6914ad2antrr 717 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴𝑌)
70 simprrr 800 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))
71 simpl 476 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → 𝐴 ∈ (𝑓𝑢))
7271ralimi 3161 . . . . . . . . . . 11 (∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → ∀𝑢𝑡 𝐴 ∈ (𝑓𝑢))
7370, 72syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∀𝑢𝑡 𝐴 ∈ (𝑓𝑢))
74 eliin 4745 . . . . . . . . . . 11 (𝐴𝑌 → (𝐴 𝑢𝑡 (𝑓𝑢) ↔ ∀𝑢𝑡 𝐴 ∈ (𝑓𝑢)))
7569, 74syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝐴 𝑢𝑡 (𝑓𝑢) ↔ ∀𝑢𝑡 𝐴 ∈ (𝑓𝑢)))
7673, 75mpbird 249 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴 𝑢𝑡 (𝑓𝑢))
77 fniinfv 6504 . . . . . . . . . 10 (𝑓 Fn 𝑡 𝑢𝑡 (𝑓𝑢) = ran 𝑓)
7853, 77syl 17 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑢𝑡 (𝑓𝑢) = ran 𝑓)
7976, 78eleqtrd 2908 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴 ran 𝑓)
8069, 79elind 4025 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴 ∈ (𝑌 ran 𝑓))
81 simprl 787 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑋 = 𝑡)
82 uniiun 4793 . . . . . . . . . . 11 𝑡 = 𝑢𝑡 𝑢
8381, 82syl6eq 2877 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑋 = 𝑢𝑡 𝑢)
8483xpeq1d 5371 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ran 𝑓)) = ( 𝑢𝑡 𝑢 × (𝑌 ran 𝑓)))
85 xpiundir 5407 . . . . . . . . 9 ( 𝑢𝑡 𝑢 × (𝑌 ran 𝑓)) = 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓))
8684, 85syl6eq 2877 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ran 𝑓)) = 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)))
87 simpr 479 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → (𝑢 × (𝑓𝑢)) ⊆ 𝑈)
8887ralimi 3161 . . . . . . . . . . 11 (∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → ∀𝑢𝑡 (𝑢 × (𝑓𝑢)) ⊆ 𝑈)
8970, 88syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∀𝑢𝑡 (𝑢 × (𝑓𝑢)) ⊆ 𝑈)
90 inss2 4058 . . . . . . . . . . . . 13 (𝑌 ran 𝑓) ⊆ ran 𝑓
9177adantr 474 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑡𝑢𝑡) → 𝑢𝑡 (𝑓𝑢) = ran 𝑓)
92 iinss2 4792 . . . . . . . . . . . . . . 15 (𝑢𝑡 𝑢𝑡 (𝑓𝑢) ⊆ (𝑓𝑢))
9392adantl 475 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑡𝑢𝑡) → 𝑢𝑡 (𝑓𝑢) ⊆ (𝑓𝑢))
9491, 93eqsstr3d 3865 . . . . . . . . . . . . 13 ((𝑓 Fn 𝑡𝑢𝑡) → ran 𝑓 ⊆ (𝑓𝑢))
9590, 94syl5ss 3838 . . . . . . . . . . . 12 ((𝑓 Fn 𝑡𝑢𝑡) → (𝑌 ran 𝑓) ⊆ (𝑓𝑢))
96 xpss2 5362 . . . . . . . . . . . 12 ((𝑌 ran 𝑓) ⊆ (𝑓𝑢) → (𝑢 × (𝑌 ran 𝑓)) ⊆ (𝑢 × (𝑓𝑢)))
97 sstr2 3834 . . . . . . . . . . . 12 ((𝑢 × (𝑌 ran 𝑓)) ⊆ (𝑢 × (𝑓𝑢)) → ((𝑢 × (𝑓𝑢)) ⊆ 𝑈 → (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈))
9895, 96, 973syl 18 . . . . . . . . . . 11 ((𝑓 Fn 𝑡𝑢𝑡) → ((𝑢 × (𝑓𝑢)) ⊆ 𝑈 → (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈))
9998ralimdva 3171 . . . . . . . . . 10 (𝑓 Fn 𝑡 → (∀𝑢𝑡 (𝑢 × (𝑓𝑢)) ⊆ 𝑈 → ∀𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈))
10053, 89, 99sylc 65 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∀𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈)
101 iunss 4781 . . . . . . . . 9 ( 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈 ↔ ∀𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈)
102100, 101sylibr 226 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈)
10386, 102eqsstrd 3864 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈)
104 eleq2 2895 . . . . . . . . 9 (𝑢 = (𝑌 ran 𝑓) → (𝐴𝑢𝐴 ∈ (𝑌 ran 𝑓)))
105 xpeq2 5363 . . . . . . . . . 10 (𝑢 = (𝑌 ran 𝑓) → (𝑋 × 𝑢) = (𝑋 × (𝑌 ran 𝑓)))
106105sseq1d 3857 . . . . . . . . 9 (𝑢 = (𝑌 ran 𝑓) → ((𝑋 × 𝑢) ⊆ 𝑈 ↔ (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈))
107104, 106anbi12d 624 . . . . . . . 8 (𝑢 = (𝑌 ran 𝑓) → ((𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈) ↔ (𝐴 ∈ (𝑌 ran 𝑓) ∧ (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈)))
108107rspcev 3526 . . . . . . 7 (((𝑌 ran 𝑓) ∈ 𝑆 ∧ (𝐴 ∈ (𝑌 ran 𝑓) ∧ (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈)) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
10968, 80, 103, 108syl12anc 870 . . . . . 6 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
110109expr 450 . . . . 5 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → ((𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
111110exlimdv 2032 . . . 4 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → (∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
112111expimpd 447 . . 3 ((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) → ((𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
113112rexlimdva 3240 . 2 (𝜑 → (∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
11437, 113mpd 15 1 (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wex 1878  wcel 2164  wne 2999  wral 3117  wrex 3118  cin 3797  wss 3798  c0 4144  𝒫 cpw 4378  {csn 4397  cop 4403   cuni 4658   cint 4697   ciun 4740   ciin 4741   × cxp 5340  ran crn 5343   Fn wfn 6118  wf 6119  ontowfo 6121  cfv 6123  (class class class)co 6905  Fincfn 8222  Topctop 21068  Compccmp 21560   ×t ctx 21734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-en 8223  df-dom 8224  df-fin 8226  df-topgen 16457  df-top 21069  df-cmp 21561  df-tx 21736
This theorem is referenced by:  txcmplem1  21815  xkoinjcn  21861  cvmlift2lem12  31831
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