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Theorem txcmplem1 21727
Description: Lemma for txcmp 21729. (Contributed by Mario Carneiro, 14-Sep-2014.)
Hypotheses
Ref Expression
txcmp.x 𝑋 = 𝑅
txcmp.y 𝑌 = 𝑆
txcmp.r (𝜑𝑅 ∈ Comp)
txcmp.s (𝜑𝑆 ∈ Comp)
txcmp.w (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
txcmp.u (𝜑 → (𝑋 × 𝑌) = 𝑊)
txcmp.a (𝜑𝐴𝑌)
Assertion
Ref Expression
txcmplem1 (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
Distinct variable groups:   𝑢,𝐴   𝑣,𝑢,𝑆   𝑢,𝑌,𝑣   𝑢,𝑊,𝑣   𝑢,𝑋,𝑣   𝜑,𝑢   𝑢,𝑅
Allowed substitution hints:   𝜑(𝑣)   𝐴(𝑣)   𝑅(𝑣)

Proof of Theorem txcmplem1
Dummy variables 𝑓 𝑘 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 txcmp.r . . 3 (𝜑𝑅 ∈ Comp)
2 id 22 . . . . . . . . 9 (𝑥𝑋𝑥𝑋)
3 txcmp.a . . . . . . . . 9 (𝜑𝐴𝑌)
4 opelxpi 5316 . . . . . . . . 9 ((𝑥𝑋𝐴𝑌) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
52, 3, 4syl2anr 590 . . . . . . . 8 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
6 txcmp.u . . . . . . . . 9 (𝜑 → (𝑋 × 𝑌) = 𝑊)
76adantr 472 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝑋 × 𝑌) = 𝑊)
85, 7eleqtrd 2846 . . . . . . 7 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑊)
9 eluni2 4600 . . . . . . 7 (⟨𝑥, 𝐴⟩ ∈ 𝑊 ↔ ∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘)
108, 9sylib 209 . . . . . 6 ((𝜑𝑥𝑋) → ∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘)
11 txcmp.w . . . . . . . . . . . 12 (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
1211adantr 472 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝑊 ⊆ (𝑅 ×t 𝑆))
1312sselda 3763 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → 𝑘 ∈ (𝑅 ×t 𝑆))
14 txcmp.s . . . . . . . . . . . . 13 (𝜑𝑆 ∈ Comp)
15 eltx 21654 . . . . . . . . . . . . 13 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
161, 14, 15syl2anc 579 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
1716adantr 472 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
1817biimpa 468 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ (𝑅 ×t 𝑆)) → ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
1913, 18syldan 585 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
20 eleq1 2832 . . . . . . . . . . . 12 (𝑦 = ⟨𝑥, 𝐴⟩ → (𝑦 ∈ (𝑟 × 𝑠) ↔ ⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠)))
2120anbi1d 623 . . . . . . . . . . 11 (𝑦 = ⟨𝑥, 𝐴⟩ → ((𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
22212rexbidv 3204 . . . . . . . . . 10 (𝑦 = ⟨𝑥, 𝐴⟩ → (∃𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
2322rspccv 3459 . . . . . . . . 9 (∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (⟨𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
2419, 23syl 17 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → (⟨𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
25 opelxp1 5320 . . . . . . . . . . . . 13 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝑥𝑟)
2625ad2antrl 719 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝑥𝑟)
27 opelxp2 5321 . . . . . . . . . . . . . . . 16 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝐴𝑠)
2827ad2antrl 719 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝐴𝑠)
2928snssd 4496 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → {𝐴} ⊆ 𝑠)
30 xpss2 5299 . . . . . . . . . . . . . 14 ({𝐴} ⊆ 𝑠 → (𝑟 × {𝐴}) ⊆ (𝑟 × 𝑠))
3129, 30syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × {𝐴}) ⊆ (𝑟 × 𝑠))
32 simprr 789 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × 𝑠) ⊆ 𝑘)
3331, 32sstrd 3773 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × {𝐴}) ⊆ 𝑘)
3426, 33jca 507 . . . . . . . . . . 11 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
3534ex 401 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → ((⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
3635rexlimdvw 3181 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → (∃𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
3736reximdv 3162 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → (∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → ∃𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
3824, 37syld 47 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → (⟨𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
3938reximdva 3163 . . . . . 6 ((𝜑𝑥𝑋) → (∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
4010, 39mpd 15 . . . . 5 ((𝜑𝑥𝑋) → ∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
41 rexcom 3246 . . . . . 6 (∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
42 r19.42v 3239 . . . . . . 7 (∃𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4342rexbii 3188 . . . . . 6 (∃𝑟𝑅𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4441, 43bitri 266 . . . . 5 (∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4540, 44sylib 209 . . . 4 ((𝜑𝑥𝑋) → ∃𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4645ralrimiva 3113 . . 3 (𝜑 → ∀𝑥𝑋𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
47 txcmp.x . . . 4 𝑋 = 𝑅
48 sseq2 3789 . . . 4 (𝑘 = (𝑓𝑟) → ((𝑟 × {𝐴}) ⊆ 𝑘 ↔ (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))
4947, 48cmpcovf 21477 . . 3 ((𝑅 ∈ Comp ∧ ∀𝑥𝑋𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘)) → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))))
501, 46, 49syl2anc 579 . 2 (𝜑 → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))))
51 txcmp.y . . . . . . . 8 𝑌 = 𝑆
521ad2antrr 717 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑅 ∈ Comp)
53 cmptop 21481 . . . . . . . . . 10 (𝑆 ∈ Comp → 𝑆 ∈ Top)
5414, 53syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ Top)
5554ad2antrr 717 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑆 ∈ Top)
56 cmptop 21481 . . . . . . . . . . 11 (𝑅 ∈ Comp → 𝑅 ∈ Top)
5752, 56syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑅 ∈ Top)
58 txtop 21655 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
5957, 55, 58syl2anc 579 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑅 ×t 𝑆) ∈ Top)
60 simprrl 799 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑓:𝑡𝑊)
6160frnd 6232 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓𝑊)
6211ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑊 ⊆ (𝑅 ×t 𝑆))
6361, 62sstrd 3773 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ⊆ (𝑅 ×t 𝑆))
64 uniopn 20984 . . . . . . . . 9 (((𝑅 ×t 𝑆) ∈ Top ∧ ran 𝑓 ⊆ (𝑅 ×t 𝑆)) → ran 𝑓 ∈ (𝑅 ×t 𝑆))
6559, 63, 64syl2anc 579 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ (𝑅 ×t 𝑆))
66 simprrr 800 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))
67 ss2iun 4694 . . . . . . . . . 10 (∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟) → 𝑟𝑡 (𝑟 × {𝐴}) ⊆ 𝑟𝑡 (𝑓𝑟))
6866, 67syl 17 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑟𝑡 (𝑟 × {𝐴}) ⊆ 𝑟𝑡 (𝑓𝑟))
69 simprl 787 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑋 = 𝑡)
70 uniiun 4731 . . . . . . . . . . . 12 𝑡 = 𝑟𝑡 𝑟
7169, 70syl6eq 2815 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑋 = 𝑟𝑡 𝑟)
7271xpeq1d 5308 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑋 × {𝐴}) = ( 𝑟𝑡 𝑟 × {𝐴}))
73 xpiundir 5344 . . . . . . . . . 10 ( 𝑟𝑡 𝑟 × {𝐴}) = 𝑟𝑡 (𝑟 × {𝐴})
7472, 73syl6req 2816 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑟𝑡 (𝑟 × {𝐴}) = (𝑋 × {𝐴}))
7560ffnd 6226 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑓 Fn 𝑡)
76 fniunfv 6699 . . . . . . . . . 10 (𝑓 Fn 𝑡 𝑟𝑡 (𝑓𝑟) = ran 𝑓)
7775, 76syl 17 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑟𝑡 (𝑓𝑟) = ran 𝑓)
7868, 74, 773sstr3d 3809 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑋 × {𝐴}) ⊆ ran 𝑓)
793ad2antrr 717 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝐴𝑌)
8047, 51, 52, 55, 65, 78, 79txtube 21726 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ ran 𝑓))
81 vex 3353 . . . . . . . . . . . . . 14 𝑓 ∈ V
8281rnex 7300 . . . . . . . . . . . . 13 ran 𝑓 ∈ V
8382elpw 4323 . . . . . . . . . . . 12 (ran 𝑓 ∈ 𝒫 𝑊 ↔ ran 𝑓𝑊)
8461, 83sylibr 225 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ 𝒫 𝑊)
85 inss2 3995 . . . . . . . . . . . . 13 (𝒫 𝑅 ∩ Fin) ⊆ Fin
86 simplr 785 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑡 ∈ (𝒫 𝑅 ∩ Fin))
8785, 86sseldi 3761 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑡 ∈ Fin)
88 dffn4 6306 . . . . . . . . . . . . 13 (𝑓 Fn 𝑡𝑓:𝑡onto→ran 𝑓)
8975, 88sylib 209 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑓:𝑡onto→ran 𝑓)
90 fofi 8461 . . . . . . . . . . . 12 ((𝑡 ∈ Fin ∧ 𝑓:𝑡onto→ran 𝑓) → ran 𝑓 ∈ Fin)
9187, 89, 90syl2anc 579 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ Fin)
9284, 91elind 3962 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin))
93 unieq 4604 . . . . . . . . . . . . 13 (𝑣 = ran 𝑓 𝑣 = ran 𝑓)
9493sseq2d 3795 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 → ((𝑋 × 𝑢) ⊆ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ran 𝑓))
9594rspcev 3462 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑢) ⊆ ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)
9695ex 401 . . . . . . . . . 10 (ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin) → ((𝑋 × 𝑢) ⊆ ran 𝑓 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
9792, 96syl 17 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ((𝑋 × 𝑢) ⊆ ran 𝑓 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
9897anim2d 605 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ((𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ ran 𝑓) → (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
9998reximdv 3162 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ ran 𝑓) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
10080, 99mpd 15 . . . . . 6 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
101100expr 448 . . . . 5 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → ((𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
102101exlimdv 2028 . . . 4 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → (∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
103102expimpd 445 . . 3 ((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) → ((𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
104103rexlimdva 3178 . 2 (𝜑 → (∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
10550, 104mpd 15 1 (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  cin 3733  wss 3734  𝒫 cpw 4317  {csn 4336  cop 4342   cuni 4596   ciun 4678   × cxp 5277  ran crn 5280   Fn wfn 6065  wf 6066  ontowfo 6068  cfv 6070  (class class class)co 6844  Fincfn 8162  Topctop 20980  Compccmp 21472   ×t ctx 21646
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 2070  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 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
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 2063  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 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-en 8163  df-dom 8164  df-fin 8166  df-topgen 16373  df-top 20981  df-bases 21033  df-cmp 21473  df-tx 21648
This theorem is referenced by:  txcmplem2  21728
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