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Theorem txcmplem2 23637
Description: Lemma for txcmp 23638. (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 479 . . . . 5 ((𝜑𝑥𝑌) → 𝑅 ∈ Comp)
61adantr 479 . . . . 5 ((𝜑𝑥𝑌) → 𝑆 ∈ Comp)
7 txcmp.w . . . . . 6 (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
87adantr 479 . . . . 5 ((𝜑𝑥𝑌) → 𝑊 ⊆ (𝑅 ×t 𝑆))
9 txcmp.u . . . . . 6 (𝜑 → (𝑋 × 𝑌) = 𝑊)
109adantr 479 . . . . 5 ((𝜑𝑥𝑌) → (𝑋 × 𝑌) = 𝑊)
11 simpr 483 . . . . 5 ((𝜑𝑥𝑌) → 𝑥𝑌)
122, 3, 5, 6, 8, 10, 11txcmplem1 23636 . . . 4 ((𝜑𝑥𝑌) → ∃𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
1312ralrimiva 3136 . . 3 (𝜑 → ∀𝑥𝑌𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
14 unieq 4924 . . . . 5 (𝑣 = (𝑓𝑢) → 𝑣 = (𝑓𝑢))
1514sseq2d 4012 . . . 4 (𝑣 = (𝑓𝑢) → ((𝑋 × 𝑢) ⊆ 𝑣 ↔ (𝑋 × 𝑢) ⊆ (𝑓𝑢)))
163, 15cmpcovf 23386 . . 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 582 . 2 (𝜑 → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))))
18 simprrl 779 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin))
19 ffn 6728 . . . . . . . . . . 11 (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤)
20 fniunfv 7262 . . . . . . . . . . 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)
2218frnd 6736 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
23 inss1 4230 . . . . . . . . . . . 12 (𝒫 𝑊 ∩ Fin) ⊆ 𝒫 𝑊
2422, 23sstrdi 3992 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ 𝒫 𝑊)
25 sspwuni 5108 . . . . . . . . . . 11 (ran 𝑓 ⊆ 𝒫 𝑊 ran 𝑓𝑊)
2624, 25sylib 217 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓𝑊)
2721, 26eqsstrd 4018 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ⊆ 𝑊)
28 vex 3466 . . . . . . . . . . 11 𝑤 ∈ V
29 fvex 6914 . . . . . . . . . . 11 (𝑓𝑧) ∈ V
3028, 29iunex 7982 . . . . . . . . . 10 𝑧𝑤 (𝑓𝑧) ∈ V
3130elpw 4611 . . . . . . . . 9 ( 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊 𝑧𝑤 (𝑓𝑧) ⊆ 𝑊)
3227, 31sylibr 233 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊)
33 inss2 4231 . . . . . . . . . 10 (𝒫 𝑆 ∩ Fin) ⊆ Fin
34 simplr 767 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ (𝒫 𝑆 ∩ Fin))
3533, 34sselid 3977 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ Fin)
36 inss2 4231 . . . . . . . . . . 11 (𝒫 𝑊 ∩ Fin) ⊆ Fin
37 fss 6744 . . . . . . . . . . 11 ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ (𝒫 𝑊 ∩ Fin) ⊆ Fin) → 𝑓:𝑤⟶Fin)
3818, 36, 37sylancl 584 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶Fin)
39 ffvelcdm 7095 . . . . . . . . . . 11 ((𝑓:𝑤⟶Fin ∧ 𝑧𝑤) → (𝑓𝑧) ∈ Fin)
4039ralrimiva 3136 . . . . . . . . . 10 (𝑓:𝑤⟶Fin → ∀𝑧𝑤 (𝑓𝑧) ∈ Fin)
4138, 40syl 17 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑓𝑧) ∈ Fin)
42 iunfi 9385 . . . . . . . . 9 ((𝑤 ∈ Fin ∧ ∀𝑧𝑤 (𝑓𝑧) ∈ Fin) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4335, 41, 42syl2anc 582 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4432, 43elind 4195 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
45 simprl 769 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑤)
46 uniiun 5066 . . . . . . . . . . . . 13 𝑤 = 𝑧𝑤 𝑧
4745, 46eqtrdi 2782 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑧𝑤 𝑧)
4847xpeq2d 5712 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = (𝑋 × 𝑧𝑤 𝑧))
49 xpiundi 5752 . . . . . . . . . . 11 (𝑋 × 𝑧𝑤 𝑧) = 𝑧𝑤 (𝑋 × 𝑧)
5048, 49eqtrdi 2782 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑋 × 𝑧))
51 simprrr 780 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))
52 xpeq2 5703 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧))
53 fveq2 6901 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
5453unieqd 4926 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 (𝑓𝑢) = (𝑓𝑧))
5552, 54sseq12d 4013 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ (𝑋 × 𝑧) ⊆ (𝑓𝑧)))
5655cbvralvw 3225 . . . . . . . . . . . 12 (∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
5751, 56sylib 217 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
58 ss2iun 5019 . . . . . . . . . . 11 (∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧) → 𝑧𝑤 (𝑋 × 𝑧) ⊆ 𝑧𝑤 (𝑓𝑧))
5957, 58syl 17 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑋 × 𝑧) ⊆ 𝑧𝑤 (𝑓𝑧))
6050, 59eqsstrd 4018 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) ⊆ 𝑧𝑤 (𝑓𝑧))
6118ffvelcdmda 7098 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
6223, 61sselid 3977 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ 𝒫 𝑊)
63 elpwi 4614 . . . . . . . . . . . . 13 ((𝑓𝑧) ∈ 𝒫 𝑊 → (𝑓𝑧) ⊆ 𝑊)
64 uniss 4921 . . . . . . . . . . . . 13 ((𝑓𝑧) ⊆ 𝑊 (𝑓𝑧) ⊆ 𝑊)
6562, 63, 643syl 18 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ⊆ 𝑊)
669ad3antrrr 728 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑋 × 𝑌) = 𝑊)
6765, 66sseqtrrd 4021 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ⊆ (𝑋 × 𝑌))
6867ralrimiva 3136 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
69 iunss 5053 . . . . . . . . . 10 ( 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌) ↔ ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7068, 69sylibr 233 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7160, 70eqssd 3997 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
72 iuncom4 5009 . . . . . . . 8 𝑧𝑤 (𝑓𝑧) = 𝑧𝑤 (𝑓𝑧)
7371, 72eqtrdi 2782 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
74 unieq 4924 . . . . . . . 8 (𝑣 = 𝑧𝑤 (𝑓𝑧) → 𝑣 = 𝑧𝑤 (𝑓𝑧))
7574rspceeqv 3630 . . . . . . 7 (( 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7644, 73, 75syl2anc 582 . . . . . 6 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7776expr 455 . . . . 5 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = 𝑤) → ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
7877exlimdv 1929 . . . 4 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = 𝑤) → (∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
7978expimpd 452 . . 3 ((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) → ((𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8079rexlimdva 3145 . 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))
8117, 80mpd 15 1 (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wex 1774  wcel 2099  wral 3051  wrex 3060  cin 3946  wss 3947  𝒫 cpw 4607   cuni 4913   ciun 5001   × cxp 5680  ran crn 5683   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  Fincfn 8974  Compccmp 23381   ×t ctx 23555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-1o 8496  df-2o 8497  df-en 8975  df-dom 8976  df-fin 8978  df-topgen 17458  df-top 22887  df-bases 22940  df-cmp 23382  df-tx 23557
This theorem is referenced by:  txcmp  23638
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