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Theorem txcmplem2 21643
Description: Lemma for txcmp 21644. (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)
Assertion
Ref Expression
txcmplem2 (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
Distinct variable groups:   𝑣,𝑆   𝑣,𝑌   𝑣,𝑊   𝑣,𝑋
Allowed substitution hints:   𝜑(𝑣)   𝑅(𝑣)

Proof of Theorem txcmplem2
Dummy variables 𝑓 𝑢 𝑥 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 txcmp.s . . 3 (𝜑𝑆 ∈ Comp)
2 txcmp.x . . . . 5 𝑋 = 𝑅
3 txcmp.y . . . . 5 𝑌 = 𝑆
4 txcmp.r . . . . . 6 (𝜑𝑅 ∈ Comp)
54adantr 472 . . . . 5 ((𝜑𝑥𝑌) → 𝑅 ∈ Comp)
61adantr 472 . . . . 5 ((𝜑𝑥𝑌) → 𝑆 ∈ Comp)
7 txcmp.w . . . . . 6 (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
87adantr 472 . . . . 5 ((𝜑𝑥𝑌) → 𝑊 ⊆ (𝑅 ×t 𝑆))
9 txcmp.u . . . . . 6 (𝜑 → (𝑋 × 𝑌) = 𝑊)
109adantr 472 . . . . 5 ((𝜑𝑥𝑌) → (𝑋 × 𝑌) = 𝑊)
11 simpr 479 . . . . 5 ((𝜑𝑥𝑌) → 𝑥𝑌)
122, 3, 5, 6, 8, 10, 11txcmplem1 21642 . . . 4 ((𝜑𝑥𝑌) → ∃𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
1312ralrimiva 3100 . . 3 (𝜑 → ∀𝑥𝑌𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
14 unieq 4592 . . . . 5 (𝑣 = (𝑓𝑢) → 𝑣 = (𝑓𝑢))
1514sseq2d 3770 . . . 4 (𝑣 = (𝑓𝑢) → ((𝑋 × 𝑢) ⊆ 𝑣 ↔ (𝑋 × 𝑢) ⊆ (𝑓𝑢)))
163, 15cmpcovf 21392 . . 3 ((𝑆 ∈ Comp ∧ ∀𝑥𝑌𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)) → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))))
171, 13, 16syl2anc 696 . 2 (𝜑 → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))))
18 simprrl 823 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin))
19 ffn 6202 . . . . . . . . . . 11 (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤)
20 fniunfv 6664 . . . . . . . . . . 11 (𝑓 Fn 𝑤 𝑧𝑤 (𝑓𝑧) = ran 𝑓)
2118, 19, 203syl 18 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) = ran 𝑓)
22 frn 6210 . . . . . . . . . . . . 13 (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
2318, 22syl 17 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
24 inss1 3972 . . . . . . . . . . . 12 (𝒫 𝑊 ∩ Fin) ⊆ 𝒫 𝑊
2523, 24syl6ss 3752 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ 𝒫 𝑊)
26 sspwuni 4759 . . . . . . . . . . 11 (ran 𝑓 ⊆ 𝒫 𝑊 ran 𝑓𝑊)
2725, 26sylib 208 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓𝑊)
2821, 27eqsstrd 3776 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ⊆ 𝑊)
29 vex 3339 . . . . . . . . . . 11 𝑤 ∈ V
30 fvex 6358 . . . . . . . . . . 11 (𝑓𝑧) ∈ V
3129, 30iunex 7308 . . . . . . . . . 10 𝑧𝑤 (𝑓𝑧) ∈ V
3231elpw 4304 . . . . . . . . 9 ( 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊 𝑧𝑤 (𝑓𝑧) ⊆ 𝑊)
3328, 32sylibr 224 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊)
34 inss2 3973 . . . . . . . . . 10 (𝒫 𝑆 ∩ Fin) ⊆ Fin
35 simplr 809 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ (𝒫 𝑆 ∩ Fin))
3634, 35sseldi 3738 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ Fin)
37 inss2 3973 . . . . . . . . . . 11 (𝒫 𝑊 ∩ Fin) ⊆ Fin
38 fss 6213 . . . . . . . . . . 11 ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ (𝒫 𝑊 ∩ Fin) ⊆ Fin) → 𝑓:𝑤⟶Fin)
3918, 37, 38sylancl 697 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶Fin)
40 ffvelrn 6516 . . . . . . . . . . 11 ((𝑓:𝑤⟶Fin ∧ 𝑧𝑤) → (𝑓𝑧) ∈ Fin)
4140ralrimiva 3100 . . . . . . . . . 10 (𝑓:𝑤⟶Fin → ∀𝑧𝑤 (𝑓𝑧) ∈ Fin)
4239, 41syl 17 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑓𝑧) ∈ Fin)
43 iunfi 8415 . . . . . . . . 9 ((𝑤 ∈ Fin ∧ ∀𝑧𝑤 (𝑓𝑧) ∈ Fin) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4436, 42, 43syl2anc 696 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4533, 44elind 3937 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
46 simprl 811 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑤)
47 uniiun 4721 . . . . . . . . . . . . 13 𝑤 = 𝑧𝑤 𝑧
4846, 47syl6eq 2806 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑧𝑤 𝑧)
4948xpeq2d 5292 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = (𝑋 × 𝑧𝑤 𝑧))
50 xpiundi 5326 . . . . . . . . . . 11 (𝑋 × 𝑧𝑤 𝑧) = 𝑧𝑤 (𝑋 × 𝑧)
5149, 50syl6eq 2806 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑋 × 𝑧))
52 simprrr 824 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))
53 xpeq2 5282 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧))
54 fveq2 6348 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
5554unieqd 4594 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 (𝑓𝑢) = (𝑓𝑧))
5653, 55sseq12d 3771 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ (𝑋 × 𝑧) ⊆ (𝑓𝑧)))
5756cbvralv 3306 . . . . . . . . . . . 12 (∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
5852, 57sylib 208 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
59 ss2iun 4684 . . . . . . . . . . 11 (∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧) → 𝑧𝑤 (𝑋 × 𝑧) ⊆ 𝑧𝑤 (𝑓𝑧))
6058, 59syl 17 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑋 × 𝑧) ⊆ 𝑧𝑤 (𝑓𝑧))
6151, 60eqsstrd 3776 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) ⊆ 𝑧𝑤 (𝑓𝑧))
6218ffvelrnda 6518 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
6324, 62sseldi 3738 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ 𝒫 𝑊)
64 elpwi 4308 . . . . . . . . . . . . 13 ((𝑓𝑧) ∈ 𝒫 𝑊 → (𝑓𝑧) ⊆ 𝑊)
65 uniss 4606 . . . . . . . . . . . . 13 ((𝑓𝑧) ⊆ 𝑊 (𝑓𝑧) ⊆ 𝑊)
6663, 64, 653syl 18 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ⊆ 𝑊)
679ad3antrrr 768 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑋 × 𝑌) = 𝑊)
6866, 67sseqtr4d 3779 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ⊆ (𝑋 × 𝑌))
6968ralrimiva 3100 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
70 iunss 4709 . . . . . . . . . 10 ( 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌) ↔ ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7169, 70sylibr 224 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7261, 71eqssd 3757 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
73 iuncom4 4676 . . . . . . . 8 𝑧𝑤 (𝑓𝑧) = 𝑧𝑤 (𝑓𝑧)
7472, 73syl6eq 2806 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
75 unieq 4592 . . . . . . . . 9 (𝑣 = 𝑧𝑤 (𝑓𝑧) → 𝑣 = 𝑧𝑤 (𝑓𝑧))
7675eqeq2d 2766 . . . . . . . 8 (𝑣 = 𝑧𝑤 (𝑓𝑧) → ((𝑋 × 𝑌) = 𝑣 ↔ (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧)))
7776rspcev 3445 . . . . . . 7 (( 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7845, 74, 77syl2anc 696 . . . . . 6 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7978expr 644 . . . . 5 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = 𝑤) → ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8079exlimdv 2006 . . . 4 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = 𝑤) → (∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8180expimpd 630 . . 3 ((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) → ((𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8281rexlimdva 3165 . 2 (𝜑 → (∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8317, 82mpd 15 1 (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wex 1849  wcel 2135  wral 3046  wrex 3047  cin 3710  wss 3711  𝒫 cpw 4298   cuni 4584   ciun 4668   × cxp 5260  ran crn 5263   Fn wfn 6040  wf 6041  cfv 6045  (class class class)co 6809  Fincfn 8117  Compccmp 21387   ×t ctx 21561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-iin 4671  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-er 7907  df-en 8118  df-dom 8119  df-fin 8121  df-topgen 16302  df-top 20897  df-bases 20948  df-cmp 21388  df-tx 21563
This theorem is referenced by:  txcmp  21644
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