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Theorem txcmplem1 21666
 Description: Lemma for txcmp 21668. (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 5305 . . . . . . . . 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 2841 . . . . . . 7 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑊)
9 eluni2 4592 . . . . . . 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 3744 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → 𝑘 ∈ (𝑅 ×t 𝑆))
14 txcmp.s . . . . . . . . . . . . 13 (𝜑𝑆 ∈ Comp)
15 eltx 21593 . . . . . . . . . . . . 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 2827 . . . . . . . . . . . 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 3195 . . . . . . . . . 10 (𝑦 = ⟨𝑥, 𝐴⟩ → (∃𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
2322rspccv 3446 . . . . . . . . 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 5307 . . . . . . . . . . . . 13 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝑥𝑟)
2625ad2antrl 766 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝑥𝑟)
27 opelxp2 5308 . . . . . . . . . . . . . . . 16 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝐴𝑠)
2827ad2antrl 766 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝐴𝑠)
2928snssd 4485 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → {𝐴} ⊆ 𝑠)
30 xpss2 5285 . . . . . . . . . . . . . 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 3754 . . . . . . . . . . . 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 3172 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → (∃𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
3736reximdv 3154 . . . . . . . 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 3155 . . . . . 6 ((𝜑𝑥𝑋) → (∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
4010, 39mpd 15 . . . . 5 ((𝜑𝑥𝑋) → ∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
41 rexcom 3237 . . . . . 6 (∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
42 r19.42v 3230 . . . . . . 7 (∃𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4342rexbii 3179 . . . . . 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 3104 . . 3 (𝜑 → ∀𝑥𝑋𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
47 txcmp.x . . . 4 𝑋 = 𝑅
48 sseq2 3768 . . . 4 (𝑘 = (𝑓𝑟) → ((𝑟 × {𝐴}) ⊆ 𝑘 ↔ (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))
4947, 48cmpcovf 21416 . . 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 21420 . . . . . . . . . 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 21420 . . . . . . . . . . 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 21594 . . . . . . . . . 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 6214 . . . . . . . . . . 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 3754 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ⊆ (𝑅 ×t 𝑆))
65 uniopn 20924 . . . . . . . . 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 4688 . . . . . . . . . 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 4725 . . . . . . . . . . . 12 𝑡 = 𝑟𝑡 𝑟
7270, 71syl6eq 2810 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑋 = 𝑟𝑡 𝑟)
7372xpeq1d 5295 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑋 × {𝐴}) = ( 𝑟𝑡 𝑟 × {𝐴}))
74 xpiundir 5331 . . . . . . . . . 10 ( 𝑟𝑡 𝑟 × {𝐴}) = 𝑟𝑡 (𝑟 × {𝐴})
7573, 74syl6req 2811 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑟𝑡 (𝑟 × {𝐴}) = (𝑋 × {𝐴}))
76 ffn 6206 . . . . . . . . . . 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 6669 . . . . . . . . . 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 3788 . . . . . . . 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 21665 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ ran 𝑓))
83 vex 3343 . . . . . . . . . . . . . 14 𝑓 ∈ V
8483rnex 7266 . . . . . . . . . . . . 13 ran 𝑓 ∈ V
8584elpw 4308 . . . . . . . . . . . 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 3977 . . . . . . . . . . . . 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 3742 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑡 ∈ Fin)
90 dffn4 6283 . . . . . . . . . . . . 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 8419 . . . . . . . . . . . 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 3941 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin))
95 unieq 4596 . . . . . . . . . . . . 13 (𝑣 = ran 𝑓 𝑣 = ran 𝑓)
9695sseq2d 3774 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 → ((𝑋 × 𝑢) ⊆ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ran 𝑓))
9796rspcev 3449 . . . . . . . . . . 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 3154 . . . . . . 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 2010 . . . 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 3169 . 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 1632  ∃wex 1853   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ∩ cin 3714   ⊆ wss 3715  𝒫 cpw 4302  {csn 4321  ⟨cop 4327  ∪ cuni 4588  ∪ ciun 4672   × cxp 5264  ran crn 5267   Fn wfn 6044  ⟶wf 6045  –onto→wfo 6047  ‘cfv 6049  (class class class)co 6814  Fincfn 8123  Topctop 20920  Compccmp 21411   ×t ctx 21585 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-fin 8127  df-topgen 16326  df-top 20921  df-bases 20972  df-cmp 21412  df-tx 21587 This theorem is referenced by:  txcmplem2  21667
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