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Theorem txtube 21723
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 2832 . . . . . . . 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 3204 . . . . . 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 21651 . . . . . . . . 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 223 . . . . . . 7 (𝜑 → ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
109adantr 472 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑈𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
11 txtube.u . . . . . . . 8 (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)
1211adantr 472 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑋 × {𝐴}) ⊆ 𝑈)
13 id 22 . . . . . . . 8 (𝑥𝑋𝑥𝑋)
14 txtube.a . . . . . . . . 9 (𝜑𝐴𝑌)
15 snidg 4364 . . . . . . . . 9 (𝐴𝑌𝐴 ∈ {𝐴})
1614, 15syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
17 opelxpi 5314 . . . . . . . 8 ((𝑥𝑋𝐴 ∈ {𝐴}) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
1813, 16, 17syl2anr 590 . . . . . . 7 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × {𝐴}))
1912, 18sseldd 3762 . . . . . 6 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑈)
204, 10, 19rspcdva 3467 . . . . 5 ((𝜑𝑥𝑋) → ∃𝑢𝑅𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
21 opelxp 5313 . . . . . . . . . 10 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ↔ (𝑥𝑢𝐴𝑣))
2221anbi1i 617 . . . . . . . . 9 ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ((𝑥𝑢𝐴𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈))
23 anass 460 . . . . . . . . 9 (((𝑥𝑢𝐴𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2422, 23bitri 266 . . . . . . . 8 ((⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2524rexbii 3188 . . . . . . 7 (∃𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑣𝑆 (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
26 r19.42v 3239 . . . . . . 7 (∃𝑣𝑆 (𝑥𝑢 ∧ (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)) ↔ (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2725, 26bitri 266 . . . . . 6 (∃𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2827rexbii 3188 . . . . 5 (∃𝑢𝑅𝑣𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑈) ↔ ∃𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
2920, 28sylib 209 . . . 4 ((𝜑𝑥𝑋) → ∃𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
3029ralrimiva 3113 . . 3 (𝜑 → ∀𝑥𝑋𝑢𝑅 (𝑥𝑢 ∧ ∃𝑣𝑆 (𝐴𝑣 ∧ (𝑢 × 𝑣) ⊆ 𝑈)))
31 txtube.x . . . 4 𝑋 = 𝑅
32 eleq2 2833 . . . . 5 (𝑣 = (𝑓𝑢) → (𝐴𝑣𝐴 ∈ (𝑓𝑢)))
33 xpeq2 5298 . . . . . 6 (𝑣 = (𝑓𝑢) → (𝑢 × 𝑣) = (𝑢 × (𝑓𝑢)))
3433sseq1d 3792 . . . . 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 21474 . . 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 4673 . . . . . . . . . 10 (ran 𝑓 = ∅ → (𝑌 ran 𝑓) = 𝑌)
3938adantl 473 . . . . . . . . 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 20990 . . . . . . . . . . 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 2844 . . . . . . . 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 6230 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ran 𝑓𝑆)
4847adantr 472 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓𝑆)
49 simpr 477 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 ≠ ∅)
50 inss2 3993 . . . . . . . . . . . . . . . 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 3759 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑡 ∈ Fin)
5346ffnd 6224 . . . . . . . . . . . . . . . 16 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑓 Fn 𝑡)
54 dffn4 6304 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑡𝑓:𝑡onto→ran 𝑓)
5553, 54sylib 209 . . . . . . . . . . . . . . 15 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑓:𝑡onto→ran 𝑓)
56 fofi 8459 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 13 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓 ∈ Fin)
59 fiinopn 20985 . . . . . . . . . . . . . 14 (𝑆 ∈ Top → ((ran 𝑓𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin) → ran 𝑓𝑆))
6059imp 395 . . . . . . . . . . . . 13 ((𝑆 ∈ Top ∧ (ran 𝑓𝑆 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin)) → ran 𝑓𝑆)
6145, 48, 49, 58, 60syl13anc 1491 . . . . . . . . . . . 12 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓𝑆)
62 elssuni 4625 . . . . . . . . . . . 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 3812 . . . . . . . . . 10 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → ran 𝑓𝑌)
65 sseqin2 3979 . . . . . . . . . 10 ( ran 𝑓𝑌 ↔ (𝑌 ran 𝑓) = ran 𝑓)
6664, 65sylib 209 . . . . . . . . 9 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → (𝑌 ran 𝑓) = ran 𝑓)
6766, 61eqeltrd 2844 . . . . . . . 8 ((((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) ∧ ran 𝑓 ≠ ∅) → (𝑌 ran 𝑓) ∈ 𝑆)
6844, 67pm2.61dane 3024 . . . . . . 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 474 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → 𝐴 ∈ (𝑓𝑢))
7271ralimi 3099 . . . . . . . . . . 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 4681 . . . . . . . . . . 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 248 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴 𝑢𝑡 (𝑓𝑢))
77 fniinfv 6446 . . . . . . . . . 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 2846 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝐴 ran 𝑓)
8069, 79elind 3960 . . . . . . 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 4729 . . . . . . . . . . 11 𝑡 = 𝑢𝑡 𝑢
8381, 82syl6eq 2815 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑋 = 𝑢𝑡 𝑢)
8483xpeq1d 5306 . . . . . . . . 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 5342 . . . . . . . . 9 ( 𝑢𝑡 𝑢 × (𝑌 ran 𝑓)) = 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓))
8684, 85syl6eq 2815 . . . . . . . 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 477 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈) → (𝑢 × (𝑓𝑢)) ⊆ 𝑈)
8887ralimi 3099 . . . . . . . . . . 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 3993 . . . . . . . . . . . . 13 (𝑌 ran 𝑓) ⊆ ran 𝑓
9177adantr 472 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑡𝑢𝑡) → 𝑢𝑡 (𝑓𝑢) = ran 𝑓)
92 iinss2 4728 . . . . . . . . . . . . . . 15 (𝑢𝑡 𝑢𝑡 (𝑓𝑢) ⊆ (𝑓𝑢))
9392adantl 473 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑡𝑢𝑡) → 𝑢𝑡 (𝑓𝑢) ⊆ (𝑓𝑢))
9491, 93eqsstr3d 3800 . . . . . . . . . . . . 13 ((𝑓 Fn 𝑡𝑢𝑡) → ran 𝑓 ⊆ (𝑓𝑢))
9590, 94syl5ss 3772 . . . . . . . . . . . 12 ((𝑓 Fn 𝑡𝑢𝑡) → (𝑌 ran 𝑓) ⊆ (𝑓𝑢))
96 xpss2 5297 . . . . . . . . . . . 12 ((𝑌 ran 𝑓) ⊆ (𝑓𝑢) → (𝑢 × (𝑌 ran 𝑓)) ⊆ (𝑢 × (𝑓𝑢)))
97 sstr2 3768 . . . . . . . . . . . 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 3109 . . . . . . . . . 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 4717 . . . . . . . . 9 ( 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈 ↔ ∀𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈)
102100, 101sylibr 225 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → 𝑢𝑡 (𝑢 × (𝑌 ran 𝑓)) ⊆ 𝑈)
10386, 102eqsstrd 3799 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈)
104 eleq2 2833 . . . . . . . . 9 (𝑢 = (𝑌 ran 𝑓) → (𝐴𝑢𝐴 ∈ (𝑌 ran 𝑓)))
105 xpeq2 5298 . . . . . . . . . 10 (𝑢 = (𝑌 ran 𝑓) → (𝑋 × 𝑢) = (𝑋 × (𝑌 ran 𝑓)))
106105sseq1d 3792 . . . . . . . . 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 3461 . . . . . . 7 (((𝑌 ran 𝑓) ∈ 𝑆 ∧ (𝐴 ∈ (𝑌 ran 𝑓) ∧ (𝑋 × (𝑌 ran 𝑓)) ⊆ 𝑈)) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
10968, 80, 103, 108syl12anc 865 . . . . . 6 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
110109expr 448 . . . . 5 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → ((𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
111110exlimdv 2028 . . . 4 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → (∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈)) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
112111expimpd 445 . . 3 ((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) → ((𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑆 ∧ ∀𝑢𝑡 (𝐴 ∈ (𝑓𝑢) ∧ (𝑢 × (𝑓𝑢)) ⊆ 𝑈))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈)))
113112rexlimdva 3178 . 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 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334  cop 4340   cuni 4594   cint 4633   ciun 4676   ciin 4677   × cxp 5275  ran crn 5278   Fn wfn 6063  wf 6064  ontowfo 6066  cfv 6068  (class class class)co 6842  Fincfn 8160  Topctop 20977  Compccmp 21469   ×t ctx 21643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-fin 8164  df-topgen 16372  df-top 20978  df-cmp 21470  df-tx 21645
This theorem is referenced by:  txcmplem1  21724  xkoinjcn  21770  cvmlift2lem12  31676
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