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Theorem txcmplem1 22992
Description: Lemma for txcmp 22994. (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 5670 . . . . . . . . 9 ((𝑥𝑋𝐴𝑌) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
52, 3, 4syl2anr 597 . . . . . . . 8 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
6 txcmp.u . . . . . . . . 9 (𝜑 → (𝑋 × 𝑌) = 𝑊)
76adantr 481 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝑋 × 𝑌) = 𝑊)
85, 7eleqtrd 2840 . . . . . . 7 ((𝜑𝑥𝑋) → ⟨𝑥, 𝐴⟩ ∈ 𝑊)
9 eluni2 4869 . . . . . . 7 (⟨𝑥, 𝐴⟩ ∈ 𝑊 ↔ ∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘)
108, 9sylib 217 . . . . . 6 ((𝜑𝑥𝑋) → ∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘)
11 txcmp.w . . . . . . . . . . . 12 (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
1211adantr 481 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝑊 ⊆ (𝑅 ×t 𝑆))
1312sselda 3944 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → 𝑘 ∈ (𝑅 ×t 𝑆))
14 txcmp.s . . . . . . . . . . . . 13 (𝜑𝑆 ∈ Comp)
15 eltx 22919 . . . . . . . . . . . . 13 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
161, 14, 15syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
1716adantr 481 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
1817biimpa 477 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ (𝑅 ×t 𝑆)) → ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
1913, 18syldan 591 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → ∀𝑦𝑘𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
20 eleq1 2825 . . . . . . . . . . . 12 (𝑦 = ⟨𝑥, 𝐴⟩ → (𝑦 ∈ (𝑟 × 𝑠) ↔ ⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠)))
2120anbi1d 630 . . . . . . . . . . 11 (𝑦 = ⟨𝑥, 𝐴⟩ → ((𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
22212rexbidv 3213 . . . . . . . . . 10 (𝑦 = ⟨𝑥, 𝐴⟩ → (∃𝑟𝑅𝑠𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ ∃𝑟𝑅𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
2322rspccv 3578 . . . . . . . . 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 5674 . . . . . . . . . . . . 13 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝑥𝑟)
2625ad2antrl 726 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝑥𝑟)
27 opelxp2 5675 . . . . . . . . . . . . . . . 16 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝐴𝑠)
2827ad2antrl 726 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝐴𝑠)
2928snssd 4769 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → {𝐴} ⊆ 𝑠)
30 xpss2 5653 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × 𝑠) ⊆ 𝑘)
3331, 32sstrd 3954 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × {𝐴}) ⊆ 𝑘)
3426, 33jca 512 . . . . . . . . . . 11 ((((𝜑𝑥𝑋) ∧ 𝑘𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
3534ex 413 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → ((⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
3635rexlimdvw 3157 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑘𝑊) → (∃𝑠𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
3736reximdv 3167 . . . . . . . 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 3165 . . . . . 6 ((𝜑𝑥𝑋) → (∃𝑘𝑊𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
4010, 39mpd 15 . . . . 5 ((𝜑𝑥𝑋) → ∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
41 rexcom 3273 . . . . . 6 (∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
42 r19.42v 3187 . . . . . . 7 (∃𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4342rexbii 3097 . . . . . 6 (∃𝑟𝑅𝑘𝑊 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4441, 43bitri 274 . . . . 5 (∃𝑘𝑊𝑟𝑅 (𝑥𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4540, 44sylib 217 . . . 4 ((𝜑𝑥𝑋) → ∃𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
4645ralrimiva 3143 . . 3 (𝜑 → ∀𝑥𝑋𝑟𝑅 (𝑥𝑟 ∧ ∃𝑘𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
47 txcmp.x . . . 4 𝑋 = 𝑅
48 sseq2 3970 . . . 4 (𝑘 = (𝑓𝑟) → ((𝑟 × {𝐴}) ⊆ 𝑘 ↔ (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))
4947, 48cmpcovf 22742 . . 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 584 . 2 (𝜑 → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))))
51 txcmp.y . . . . . . . 8 𝑌 = 𝑆
521ad2antrr 724 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑅 ∈ Comp)
53 cmptop 22746 . . . . . . . . . 10 (𝑆 ∈ Comp → 𝑆 ∈ Top)
5414, 53syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ Top)
5554ad2antrr 724 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑆 ∈ Top)
56 cmptop 22746 . . . . . . . . . . 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 22920 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
5957, 55, 58syl2anc 584 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑅 ×t 𝑆) ∈ Top)
60 simprrl 779 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑓:𝑡𝑊)
6160frnd 6676 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓𝑊)
6211ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑊 ⊆ (𝑅 ×t 𝑆))
6361, 62sstrd 3954 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ⊆ (𝑅 ×t 𝑆))
64 uniopn 22246 . . . . . . . . 9 (((𝑅 ×t 𝑆) ∈ Top ∧ ran 𝑓 ⊆ (𝑅 ×t 𝑆)) → ran 𝑓 ∈ (𝑅 ×t 𝑆))
6559, 63, 64syl2anc 584 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ (𝑅 ×t 𝑆))
66 simprrr 780 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))
67 ss2iun 4972 . . . . . . . . . 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 769 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑋 = 𝑡)
70 uniiun 5018 . . . . . . . . . . . 12 𝑡 = 𝑟𝑡 𝑟
7169, 70eqtrdi 2792 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑋 = 𝑟𝑡 𝑟)
7271xpeq1d 5662 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑋 × {𝐴}) = ( 𝑟𝑡 𝑟 × {𝐴}))
73 xpiundir 5703 . . . . . . . . . 10 ( 𝑟𝑡 𝑟 × {𝐴}) = 𝑟𝑡 (𝑟 × {𝐴})
7472, 73eqtr2di 2793 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑟𝑡 (𝑟 × {𝐴}) = (𝑋 × {𝐴}))
7560ffnd 6669 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑓 Fn 𝑡)
76 fniunfv 7194 . . . . . . . . . 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 3990 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → (𝑋 × {𝐴}) ⊆ ran 𝑓)
793ad2antrr 724 . . . . . . . 8 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝐴𝑌)
8047, 51, 52, 55, 65, 78, 79txtube 22991 . . . . . . 7 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ ran 𝑓))
81 vex 3449 . . . . . . . . . . . . . 14 𝑓 ∈ V
8281rnex 7849 . . . . . . . . . . . . 13 ran 𝑓 ∈ V
8382elpw 4564 . . . . . . . . . . . 12 (ran 𝑓 ∈ 𝒫 𝑊 ↔ ran 𝑓𝑊)
8461, 83sylibr 233 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ 𝒫 𝑊)
85 simplr 767 . . . . . . . . . . . . 13 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑡 ∈ (𝒫 𝑅 ∩ Fin))
8685elin2d 4159 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑡 ∈ Fin)
87 dffn4 6762 . . . . . . . . . . . . 13 (𝑓 Fn 𝑡𝑓:𝑡onto→ran 𝑓)
8875, 87sylib 217 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → 𝑓:𝑡onto→ran 𝑓)
89 fofi 9282 . . . . . . . . . . . 12 ((𝑡 ∈ Fin ∧ 𝑓:𝑡onto→ran 𝑓) → ran 𝑓 ∈ Fin)
9086, 88, 89syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ Fin)
9184, 90elind 4154 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin))
92 unieq 4876 . . . . . . . . . . . . 13 (𝑣 = ran 𝑓 𝑣 = ran 𝑓)
9392sseq2d 3976 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 → ((𝑋 × 𝑢) ⊆ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ran 𝑓))
9493rspcev 3581 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑢) ⊆ ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)
9594ex 413 . . . . . . . . . 10 (ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin) → ((𝑋 × 𝑢) ⊆ ran 𝑓 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
9691, 95syl 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))
9796anim2d 612 . . . . . . . 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)))
9897reximdv 3167 . . . . . . 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)))
9980, 98mpd 15 . . . . . 6 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = 𝑡 ∧ (𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)))) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
10099expr 457 . . . . 5 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → ((𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
101100exlimdv 1936 . . . 4 (((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = 𝑡) → (∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟)) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
102101expimpd 454 . . 3 ((𝜑𝑡 ∈ (𝒫 𝑅 ∩ Fin)) → ((𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
103102rexlimdva 3152 . 2 (𝜑 → (∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = 𝑡 ∧ ∃𝑓(𝑓:𝑡𝑊 ∧ ∀𝑟𝑡 (𝑟 × {𝐴}) ⊆ (𝑓𝑟))) → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)))
10450, 103mpd 15 1 (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  cin 3909  wss 3910  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   ciun 4954   × cxp 5631  ran crn 5634   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  Fincfn 8883  Topctop 22242  Compccmp 22737   ×t ctx 22911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-fin 8887  df-topgen 17325  df-top 22243  df-bases 22296  df-cmp 22738  df-tx 22913
This theorem is referenced by:  txcmplem2  22993
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