| Step | Hyp | Ref
| Expression |
| 1 | | txcmp.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ Comp) |
| 2 | | txcmp.x |
. . . . 5
⊢ 𝑋 = ∪
𝑅 |
| 3 | | txcmp.y |
. . . . 5
⊢ 𝑌 = ∪
𝑆 |
| 4 | | txcmp.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ Comp) |
| 5 | 4 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ Comp) |
| 6 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ Comp) |
| 7 | | txcmp.w |
. . . . . 6
⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×t 𝑆)) |
| 8 | 7 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑊 ⊆ (𝑅 ×t 𝑆)) |
| 9 | | txcmp.u |
. . . . . 6
⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊) |
| 10 | 9 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑋 × 𝑌) = ∪ 𝑊) |
| 11 | | simpr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌) |
| 12 | 2, 3, 5, 6, 8, 10,
11 | txcmplem1 23649 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)) |
| 13 | 12 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)) |
| 14 | | unieq 4918 |
. . . . 5
⊢ (𝑣 = (𝑓‘𝑢) → ∪ 𝑣 = ∪
(𝑓‘𝑢)) |
| 15 | 14 | sseq2d 4016 |
. . . 4
⊢ (𝑣 = (𝑓‘𝑢) → ((𝑋 × 𝑢) ⊆ ∪ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))) |
| 16 | 3, 15 | cmpcovf 23399 |
. . 3
⊢ ((𝑆 ∈ Comp ∧ ∀𝑥 ∈ 𝑌 ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)) → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) |
| 17 | 1, 13, 16 | syl2anc 584 |
. 2
⊢ (𝜑 → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) |
| 18 | | simprrl 781 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin)) |
| 19 | | ffn 6736 |
. . . . . . . . . . 11
⊢ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤) |
| 20 | | fniunfv 7267 |
. . . . . . . . . . 11
⊢ (𝑓 Fn 𝑤 → ∪
𝑧 ∈ 𝑤 (𝑓‘𝑧) = ∪ ran 𝑓) |
| 21 | 18, 19, 20 | 3syl 18 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) = ∪ ran 𝑓) |
| 22 | 18 | frnd 6744 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin)) |
| 23 | | inss1 4237 |
. . . . . . . . . . . 12
⊢
(𝒫 𝑊 ∩
Fin) ⊆ 𝒫 𝑊 |
| 24 | 22, 23 | sstrdi 3996 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ran 𝑓 ⊆ 𝒫 𝑊) |
| 25 | | sspwuni 5100 |
. . . . . . . . . . 11
⊢ (ran
𝑓 ⊆ 𝒫 𝑊 ↔ ∪ ran 𝑓 ⊆ 𝑊) |
| 26 | 24, 25 | sylib 218 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ ran
𝑓 ⊆ 𝑊) |
| 27 | 21, 26 | eqsstrd 4018 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ⊆ 𝑊) |
| 28 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑤 ∈ V |
| 29 | | fvex 6919 |
. . . . . . . . . . 11
⊢ (𝑓‘𝑧) ∈ V |
| 30 | 28, 29 | iunex 7993 |
. . . . . . . . . 10
⊢ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ V |
| 31 | 30 | elpw 4604 |
. . . . . . . . 9
⊢ (∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ 𝒫 𝑊 ↔ ∪
𝑧 ∈ 𝑤 (𝑓‘𝑧) ⊆ 𝑊) |
| 32 | 27, 31 | sylibr 234 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ 𝒫 𝑊) |
| 33 | | inss2 4238 |
. . . . . . . . . 10
⊢
(𝒫 𝑆 ∩
Fin) ⊆ Fin |
| 34 | | simplr 769 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) |
| 35 | 33, 34 | sselid 3981 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 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) |
| 38 | 18, 36, 37 | sylancl 586 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑓:𝑤⟶Fin) |
| 39 | | ffvelcdm 7101 |
. . . . . . . . . . 11
⊢ ((𝑓:𝑤⟶Fin ∧ 𝑧 ∈ 𝑤) → (𝑓‘𝑧) ∈ Fin) |
| 40 | 39 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (𝑓:𝑤⟶Fin → ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin) |
| 41 | 38, 40 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin) |
| 42 | | iunfi 9383 |
. . . . . . . . 9
⊢ ((𝑤 ∈ Fin ∧ ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin) |
| 43 | 35, 41, 42 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin) |
| 44 | 32, 43 | elind 4200 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ (𝒫 𝑊 ∩ Fin)) |
| 45 | | simprl 771 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑌 = ∪ 𝑤) |
| 46 | | uniiun 5058 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑤 =
∪ 𝑧 ∈ 𝑤 𝑧 |
| 47 | 45, 46 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑌 = ∪ 𝑧 ∈ 𝑤 𝑧) |
| 48 | 47 | xpeq2d 5715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) = (𝑋 × ∪
𝑧 ∈ 𝑤 𝑧)) |
| 49 | | xpiundi 5756 |
. . . . . . . . . . 11
⊢ (𝑋 × ∪ 𝑧 ∈ 𝑤 𝑧) = ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧) |
| 50 | 48, 49 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) = ∪
𝑧 ∈ 𝑤 (𝑋 × 𝑧)) |
| 51 | | simprrr 782 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)) |
| 52 | | xpeq2 5706 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧)) |
| 53 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧)) |
| 54 | 53 | unieqd 4920 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = 𝑧 → ∪ (𝑓‘𝑢) = ∪ (𝑓‘𝑧)) |
| 55 | 52, 54 | sseq12d 4017 |
. . . . . . . . . . . . 13
⊢ (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢) ↔ (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧))) |
| 56 | 55 | cbvralvw 3237 |
. . . . . . . . . . . 12
⊢
(∀𝑢 ∈
𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢) ↔ ∀𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧)) |
| 57 | 51, 56 | sylib 218 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∀𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧)) |
| 58 | | ss2iun 5010 |
. . . . . . . . . . 11
⊢
(∀𝑧 ∈
𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧) → ∪
𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪
𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧)) |
| 59 | 57, 58 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪
𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧)) |
| 60 | 50, 59 | eqsstrd 4018 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) ⊆ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧)) |
| 61 | 18 | ffvelcdmda 7104 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → (𝑓‘𝑧) ∈ (𝒫 𝑊 ∩ Fin)) |
| 62 | 23, 61 | sselid 3981 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → (𝑓‘𝑧) ∈ 𝒫 𝑊) |
| 63 | | elpwi 4607 |
. . . . . . . . . . . . 13
⊢ ((𝑓‘𝑧) ∈ 𝒫 𝑊 → (𝑓‘𝑧) ⊆ 𝑊) |
| 64 | | uniss 4915 |
. . . . . . . . . . . . 13
⊢ ((𝑓‘𝑧) ⊆ 𝑊 → ∪ (𝑓‘𝑧) ⊆ ∪ 𝑊) |
| 65 | 62, 63, 64 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → ∪ (𝑓‘𝑧) ⊆ ∪ 𝑊) |
| 66 | 9 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → (𝑋 × 𝑌) = ∪ 𝑊) |
| 67 | 65, 66 | sseqtrrd 4021 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌)) |
| 68 | 67 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∀𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌)) |
| 69 | | iunss 5045 |
. . . . . . . . . 10
⊢ (∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌) ↔ ∀𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌)) |
| 70 | 68, 69 | sylibr 234 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌)) |
| 71 | 60, 70 | eqssd 4001 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) = ∪
𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧)) |
| 72 | | iuncom4 5000 |
. . . . . . . 8
⊢ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) |
| 73 | 71, 72 | eqtrdi 2793 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)) |
| 74 | | unieq 4918 |
. . . . . . . 8
⊢ (𝑣 = ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) → ∪ 𝑣 = ∪
∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)) |
| 75 | 74 | rspceeqv 3645 |
. . . . . . 7
⊢
((∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑌) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣) |
| 76 | 44, 73, 75 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣) |
| 77 | 76 | expr 456 |
. . . . 5
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = ∪ 𝑤) → ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)) |
| 78 | 77 | exlimdv 1933 |
. . . 4
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = ∪ 𝑤) → (∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)) |
| 79 | 78 | expimpd 453 |
. . 3
⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) → ((𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)) |
| 80 | 79 | rexlimdva 3155 |
. 2
⊢ (𝜑 → (∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)) |
| 81 | 17, 80 | mpd 15 |
1
⊢ (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣) |