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Theorem txcmplem2 23650
Description: Lemma for txcmp 23651. (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 480 . . . . 5 ((𝜑𝑥𝑌) → 𝑅 ∈ Comp)
61adantr 480 . . . . 5 ((𝜑𝑥𝑌) → 𝑆 ∈ Comp)
7 txcmp.w . . . . . 6 (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
87adantr 480 . . . . 5 ((𝜑𝑥𝑌) → 𝑊 ⊆ (𝑅 ×t 𝑆))
9 txcmp.u . . . . . 6 (𝜑 → (𝑋 × 𝑌) = 𝑊)
109adantr 480 . . . . 5 ((𝜑𝑥𝑌) → (𝑋 × 𝑌) = 𝑊)
11 simpr 484 . . . . 5 ((𝜑𝑥𝑌) → 𝑥𝑌)
122, 3, 5, 6, 8, 10, 11txcmplem1 23649 . . . 4 ((𝜑𝑥𝑌) → ∃𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
1312ralrimiva 3146 . . 3 (𝜑 → ∀𝑥𝑌𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
14 unieq 4918 . . . . 5 (𝑣 = (𝑓𝑢) → 𝑣 = (𝑓𝑢))
1514sseq2d 4016 . . . 4 (𝑣 = (𝑓𝑢) → ((𝑋 × 𝑢) ⊆ 𝑣 ↔ (𝑋 × 𝑢) ⊆ (𝑓𝑢)))
163, 15cmpcovf 23399 . . 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 584 . 2 (𝜑 → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))))
18 simprrl 781 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin))
19 ffn 6736 . . . . . . . . . . 11 (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤)
20 fniunfv 7267 . . . . . . . . . . 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 6744 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
23 inss1 4237 . . . . . . . . . . . 12 (𝒫 𝑊 ∩ Fin) ⊆ 𝒫 𝑊
2422, 23sstrdi 3996 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ 𝒫 𝑊)
25 sspwuni 5100 . . . . . . . . . . 11 (ran 𝑓 ⊆ 𝒫 𝑊 ran 𝑓𝑊)
2624, 25sylib 218 . . . . . . . . . 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 3484 . . . . . . . . . . 11 𝑤 ∈ V
29 fvex 6919 . . . . . . . . . . 11 (𝑓𝑧) ∈ V
3028, 29iunex 7993 . . . . . . . . . 10 𝑧𝑤 (𝑓𝑧) ∈ V
3130elpw 4604 . . . . . . . . 9 ( 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊 𝑧𝑤 (𝑓𝑧) ⊆ 𝑊)
3227, 31sylibr 234 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊)
33 inss2 4238 . . . . . . . . . 10 (𝒫 𝑆 ∩ Fin) ⊆ Fin
34 simplr 769 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ (𝒫 𝑆 ∩ Fin))
3533, 34sselid 3981 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ Fin)
36 inss2 4238 . . . . . . . . . . 11 (𝒫 𝑊 ∩ Fin) ⊆ Fin
37 fss 6752 . . . . . . . . . . 11 ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ (𝒫 𝑊 ∩ Fin) ⊆ Fin) → 𝑓:𝑤⟶Fin)
3818, 36, 37sylancl 586 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶Fin)
39 ffvelcdm 7101 . . . . . . . . . . 11 ((𝑓:𝑤⟶Fin ∧ 𝑧𝑤) → (𝑓𝑧) ∈ Fin)
4039ralrimiva 3146 . . . . . . . . . 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 9383 . . . . . . . . 9 ((𝑤 ∈ Fin ∧ ∀𝑧𝑤 (𝑓𝑧) ∈ Fin) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4335, 41, 42syl2anc 584 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4432, 43elind 4200 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
45 simprl 771 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑤)
46 uniiun 5058 . . . . . . . . . . . . 13 𝑤 = 𝑧𝑤 𝑧
4745, 46eqtrdi 2793 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑧𝑤 𝑧)
4847xpeq2d 5715 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = (𝑋 × 𝑧𝑤 𝑧))
49 xpiundi 5756 . . . . . . . . . . 11 (𝑋 × 𝑧𝑤 𝑧) = 𝑧𝑤 (𝑋 × 𝑧)
5048, 49eqtrdi 2793 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑋 × 𝑧))
51 simprrr 782 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))
52 xpeq2 5706 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧))
53 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
5453unieqd 4920 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 (𝑓𝑢) = (𝑓𝑧))
5552, 54sseq12d 4017 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ (𝑋 × 𝑧) ⊆ (𝑓𝑧)))
5655cbvralvw 3237 . . . . . . . . . . . 12 (∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
5751, 56sylib 218 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
58 ss2iun 5010 . . . . . . . . . . 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 7104 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
6223, 61sselid 3981 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ 𝒫 𝑊)
63 elpwi 4607 . . . . . . . . . . . . 13 ((𝑓𝑧) ∈ 𝒫 𝑊 → (𝑓𝑧) ⊆ 𝑊)
64 uniss 4915 . . . . . . . . . . . . 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 730 . . . . . . . . . . . 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 3146 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
69 iunss 5045 . . . . . . . . . 10 ( 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌) ↔ ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7068, 69sylibr 234 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7160, 70eqssd 4001 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
72 iuncom4 5000 . . . . . . . 8 𝑧𝑤 (𝑓𝑧) = 𝑧𝑤 (𝑓𝑧)
7371, 72eqtrdi 2793 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
74 unieq 4918 . . . . . . . 8 (𝑣 = 𝑧𝑤 (𝑓𝑧) → 𝑣 = 𝑧𝑤 (𝑓𝑧))
7574rspceeqv 3645 . . . . . . 7 (( 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7644, 73, 75syl2anc 584 . . . . . 6 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7776expr 456 . . . . 5 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = 𝑤) → ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
7877exlimdv 1933 . . . 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 453 . . 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 3155 . 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 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   ciun 4991   × cxp 5683  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  Compccmp 23394   ×t ctx 23568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-en 8986  df-dom 8987  df-fin 8989  df-topgen 17488  df-top 22900  df-bases 22953  df-cmp 23395  df-tx 23570
This theorem is referenced by:  txcmp  23651
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