MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  txcmplem1 Structured version   Visualization version   GIF version

Theorem txcmplem1 21567
Description: Lemma for txcmp 21569. (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 5257 . . . . . . . . 9 ((𝑥𝑋𝐴𝑌) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
52, 3, 4syl2anr 496 . . . . . . . 8 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
6 txcmp.u . . . . . . . . 9 (𝜑 → (𝑋 × 𝑌) = 𝑊)
76adantr 472 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝑋 × 𝑌) = 𝑊)
85, 7eleqtrd 2805 . . . . . . 7 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑊)
9 eluni2 4548 . . . . . . 7 (⟨𝑥, 𝐴⟩ ∈ 𝑊 ↔ ∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘)
108, 9sylib 208 . . . . . 6 ((𝜑𝑥𝑋) → ∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘)
11 txcmp.w . . . . . . . . . . . 12 (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
1211adantr 472 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝑊 ⊆ (𝑅 ×t 𝑆))
1312sselda 3709 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → 𝑘 ∈ (𝑅 ×t 𝑆))
14 txcmp.s . . . . . . . . . . . . 13 (𝜑𝑆 ∈ Comp)
15 eltx 21494 . . . . . . . . . . . . 13 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
161, 14, 15syl2anc 696 . . . . . . . . . . . 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 502 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ (𝑅 ×t 𝑆)) → ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
1913, 18syldan 488 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
20 eleq1 2791 . . . . . . . . . . . 12 (𝑦 = ⟨𝑥, 𝐴⟩ → (𝑦 ∈ (𝑟 × 𝑠) ↔ ⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠)))
2120anbi1d 743 . . . . . . . . . . 11 (𝑦 = ⟨𝑥, 𝐴⟩ → ((𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
22212rexbidv 3159 . . . . . . . . . 10 (𝑦 = ⟨𝑥, 𝐴⟩ → (∃𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
2322rspccv 3410 . . . . . . . . 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 5259 . . . . . . . . . . . . 13 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝑥𝑟)
2625ad2antrl 766 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝑥𝑟)
27 opelxp2 5260 . . . . . . . . . . . . . . . 16 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝐴𝑠)
2827ad2antrl 766 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝐴𝑠)
2928snssd 4448 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → {𝐴} ⊆ 𝑠)
30 xpss2 5237 . . . . . . . . . . . . . 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 813 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × 𝑠) ⊆ 𝑘)
3331, 32sstrd 3719 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × {𝐴}) ⊆ 𝑘)
3426, 33jca 555 . . . . . . . . . . 11 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
3534ex 449 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → ((⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
3635rexlimdvw 3136 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → (∃𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
3736reximdv 3118 . . . . . . . 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 3119 . . . . . 6 ((𝜑𝑥𝑋) → (∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
4010, 39mpd 15 . . . . 5 ((𝜑𝑥𝑋) → ∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
41 rexcom 3201 . . . . . 6 (∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
42 r19.42v 3194 . . . . . . 7 (∃𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4342rexbii 3143 . . . . . 6 (∃𝑟𝑅𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4441, 43bitri 264 . . . . 5 (∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4540, 44sylib 208 . . . 4 ((𝜑𝑥𝑋) → ∃𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4645ralrimiva 3068 . . 3 (𝜑 → ∀𝑥𝑋𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
47 txcmp.x . . . 4 𝑋 = 𝑅
48 sseq2 3733 . . . 4 (𝑘 = (𝑓𝑟) → ((𝑟 × {𝐴}) ⊆ 𝑘 ↔ (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))
4947, 48cmpcovf 21317 . . 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 696 . 2 (𝜑 → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))))
51 txcmp.y . . . . . . . 8 𝑌 = 𝑆
521ad2antrr 764 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑅 ∈ Comp)
53 cmptop 21321 . . . . . . . . . 10 (𝑆 ∈ Comp → 𝑆 ∈ Top)
5414, 53syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ Top)
5554ad2antrr 764 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑆 ∈ Top)
56 cmptop 21321 . . . . . . . . . . 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 21495 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
5957, 55, 58syl2anc 696 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑅 ×t 𝑆) ∈ Top)
60 simprrl 823 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑓:𝑡𝑊)
61 frn 6166 . . . . . . . . . . 11 (𝑓:𝑡𝑊 → ran 𝑓𝑊)
6260, 61syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓𝑊)
6311ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑊 ⊆ (𝑅 ×t 𝑆))
6462, 63sstrd 3719 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ⊆ (𝑅 ×t 𝑆))
65 uniopn 20825 . . . . . . . . 9 (((𝑅 ×t 𝑆) ∈ Top ∧ ran 𝑓 ⊆ (𝑅 ×t 𝑆)) → ran 𝑓 ∈ (𝑅 ×t 𝑆))
6659, 64, 65syl2anc 696 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ (𝑅 ×t 𝑆))
67 simprrr 824 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))
68 ss2iun 4644 . . . . . . . . . 10 (∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟) → 𝑟𝑡 (𝑟 × {𝐴}) ⊆ 𝑟𝑡 (𝑓𝑟))
6967, 68syl 17 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑟𝑡 (𝑟 × {𝐴}) ⊆ 𝑟𝑡 (𝑓𝑟))
70 simprl 811 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑋 = 𝑡)
71 uniiun 4681 . . . . . . . . . . . 12 𝑡 = 𝑟𝑡 𝑟
7270, 71syl6eq 2774 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑋 = 𝑟𝑡 𝑟)
7372xpeq1d 5247 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑋 × {𝐴}) = ( 𝑟𝑡 𝑟 × {𝐴}))
74 xpiundir 5283 . . . . . . . . . 10 ( 𝑟𝑡 𝑟 × {𝐴}) = 𝑟𝑡 (𝑟 × {𝐴})
7573, 74syl6req 2775 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑟𝑡 (𝑟 × {𝐴}) = (𝑋 × {𝐴}))
76 ffn 6158 . . . . . . . . . . 11 (𝑓:𝑡𝑊𝑓 Fn 𝑡)
7760, 76syl 17 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑓 Fn 𝑡)
78 fniunfv 6620 . . . . . . . . . 10 (𝑓 Fn 𝑡 𝑟𝑡 (𝑓𝑟) = ran 𝑓)
7977, 78syl 17 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑟𝑡 (𝑓𝑟) = ran 𝑓)
8069, 75, 793sstr3d 3753 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑋 × {𝐴}) ⊆ ran 𝑓)
813ad2antrr 764 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝐴𝑌)
8247, 51, 52, 55, 66, 80, 81txtube 21566 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ ran 𝑓))
83 vex 3307 . . . . . . . . . . . . . 14 𝑓 ∈ V
8483rnex 7217 . . . . . . . . . . . . 13 ran 𝑓 ∈ V
8584elpw 4272 . . . . . . . . . . . 12 (ran 𝑓 ∈ 𝒫 𝑊 ↔ ran 𝑓𝑊)
8662, 85sylibr 224 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ 𝒫 𝑊)
87 inss2 3942 . . . . . . . . . . . . 13 (𝒫 𝑅 ∩ Fin) ⊆ Fin
88 simplr 809 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑡 ∈ (𝒫 𝑅 ∩ Fin))
8987, 88sseldi 3707 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑡 ∈ Fin)
90 dffn4 6234 . . . . . . . . . . . . 13 (𝑓 Fn 𝑡𝑓:𝑡onto→ran 𝑓)
9177, 90sylib 208 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑓:𝑡onto→ran 𝑓)
92 fofi 8368 . . . . . . . . . . . 12 ((𝑡 ∈ Fin ∧ 𝑓:𝑡onto→ran 𝑓) → ran 𝑓 ∈ Fin)
9389, 91, 92syl2anc 696 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ Fin)
9486, 93elind 3906 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin))
95 unieq 4552 . . . . . . . . . . . . 13 (𝑣 = ran 𝑓 𝑣 = ran 𝑓)
9695sseq2d 3739 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 → ((𝑋 × 𝑢) ⊆ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ran 𝑓))
9796rspcev 3413 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑢) ⊆ ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)
9897ex 449 . . . . . . . . . 10 (ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin) → ((𝑋 × 𝑢) ⊆ ran 𝑓 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
9994, 98syl 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))
10099anim2d 590 . . . . . . . 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)))
101100reximdv 3118 . . . . . . 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)))
10282, 101mpd 15 . . . . . 6 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
103102expr 644 . . . . 5 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → ((𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
104103exlimdv 1974 . . . 4 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → (∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
105104expimpd 630 . . 3 ((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) → ((𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
106105rexlimdva 3133 . 2 (𝜑 → (∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
10750, 106mpd 15 1 (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wex 1817  wcel 2103  wral 3014  wrex 3015  cin 3679  wss 3680  𝒫 cpw 4266  {csn 4285  cop 4291   cuni 4544   ciun 4628   × cxp 5216  ran crn 5219   Fn wfn 5996  wf 5997  ontowfo 5999  cfv 6001  (class class class)co 6765  Fincfn 8072  Topctop 20821  Compccmp 21312   ×t ctx 21486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-fin 8076  df-topgen 16227  df-top 20822  df-bases 20873  df-cmp 21313  df-tx 21488
This theorem is referenced by:  txcmplem2  21568
  Copyright terms: Public domain W3C validator