| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | cntop2 23250 | . . 3
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | 
| 2 | 1 | 3ad2ant3 1135 | . 2
⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top) | 
| 3 |  | elpwi 4606 | . . . 4
⊢ (𝑢 ∈ 𝒫 𝐾 → 𝑢 ⊆ 𝐾) | 
| 4 |  | simpl1 1191 | . . . . . . 7
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐽 ∈ Comp) | 
| 5 |  | simpl3 1193 | . . . . . . . . . 10
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾)) | 
| 6 |  | simprl 770 | . . . . . . . . . . 11
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑢 ⊆ 𝐾) | 
| 7 | 6 | sselda 3982 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝐾) | 
| 8 |  | cnima 23274 | . . . . . . . . . 10
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽) | 
| 9 | 5, 7, 8 | syl2an2r 685 | . . . . . . . . 9
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (◡𝐹 “ 𝑦) ∈ 𝐽) | 
| 10 | 9 | fmpttd 7134 | . . . . . . . 8
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)):𝑢⟶𝐽) | 
| 11 | 10 | frnd 6743 | . . . . . . 7
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ⊆ 𝐽) | 
| 12 |  | simprr 772 | . . . . . . . . 9
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑌 = ∪ 𝑢) | 
| 13 | 12 | imaeq2d 6077 | . . . . . . . 8
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (◡𝐹 “ 𝑌) = (◡𝐹 “ ∪ 𝑢)) | 
| 14 |  | eqid 2736 | . . . . . . . . . . 11
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 15 |  | cncmp.2 | . . . . . . . . . . 11
⊢ 𝑌 = ∪
𝐾 | 
| 16 | 14, 15 | cnf 23255 | . . . . . . . . . 10
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌) | 
| 17 | 5, 16 | syl 17 | . . . . . . . . 9
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐹:∪ 𝐽⟶𝑌) | 
| 18 |  | fimacnv 6757 | . . . . . . . . 9
⊢ (𝐹:∪
𝐽⟶𝑌 → (◡𝐹 “ 𝑌) = ∪ 𝐽) | 
| 19 | 17, 18 | syl 17 | . . . . . . . 8
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (◡𝐹 “ 𝑌) = ∪ 𝐽) | 
| 20 | 9 | ralrimiva 3145 | . . . . . . . . . 10
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (◡𝐹 “ 𝑦) ∈ 𝐽) | 
| 21 |  | dfiun2g 5029 | . . . . . . . . . 10
⊢
(∀𝑦 ∈
𝑢 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪
𝑦 ∈ 𝑢 (◡𝐹 “ 𝑦) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (◡𝐹 “ 𝑦)}) | 
| 22 | 20, 21 | syl 17 | . . . . . . . . 9
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∪ 𝑦 ∈ 𝑢 (◡𝐹 “ 𝑦) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (◡𝐹 “ 𝑦)}) | 
| 23 |  | imauni 7267 | . . . . . . . . 9
⊢ (◡𝐹 “ ∪ 𝑢) = ∪ 𝑦 ∈ 𝑢 (◡𝐹 “ 𝑦) | 
| 24 |  | eqid 2736 | . . . . . . . . . . 11
⊢ (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) = (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) | 
| 25 | 24 | rnmpt 5967 | . . . . . . . . . 10
⊢ ran
(𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) = {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (◡𝐹 “ 𝑦)} | 
| 26 | 25 | unieqi 4918 | . . . . . . . . 9
⊢ ∪ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (◡𝐹 “ 𝑦)} | 
| 27 | 22, 23, 26 | 3eqtr4g 2801 | . . . . . . . 8
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (◡𝐹 “ ∪ 𝑢) = ∪
ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))) | 
| 28 | 13, 19, 27 | 3eqtr3d 2784 | . . . . . . 7
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∪ 𝐽 =
∪ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))) | 
| 29 | 14 | cmpcov 23398 | . . . . . . 7
⊢ ((𝐽 ∈ Comp ∧ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ⊆ 𝐽 ∧ ∪ 𝐽 = ∪
ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))) → ∃𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∩ Fin)∪
𝐽 = ∪ 𝑠) | 
| 30 | 4, 11, 28, 29 | syl3anc 1372 | . . . . . 6
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∃𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∩ Fin)∪
𝐽 = ∪ 𝑠) | 
| 31 |  | elfpw 9395 | . . . . . . . 8
⊢ (𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∩ Fin) ↔ (𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin)) | 
| 32 |  | simprll 778 | . . . . . . . . . . . . . . 15
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → 𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))) | 
| 33 | 32 | sselda 3982 | . . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ Comp
∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) ∧ 𝑐 ∈ 𝑠) → 𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))) | 
| 34 |  | simpll2 1213 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝐹:𝑋–onto→𝑌) | 
| 35 |  | elssuni 4936 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾) | 
| 36 | 35, 15 | sseqtrrdi 4024 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ 𝑌) | 
| 37 | 7, 36 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ⊆ 𝑌) | 
| 38 |  | foimacnv 6864 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦) | 
| 39 | 34, 37, 38 | syl2anc 584 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦) | 
| 40 |  | simpr 484 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝑢) | 
| 41 | 39, 40 | eqeltrd 2840 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝑢) | 
| 42 | 41 | ralrimiva 3145 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝑢) | 
| 43 |  | imaeq2 6073 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑐) = (𝐹 “ (◡𝐹 “ 𝑦))) | 
| 44 | 43 | eleq1d 2825 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = (◡𝐹 “ 𝑦) → ((𝐹 “ 𝑐) ∈ 𝑢 ↔ (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝑢)) | 
| 45 | 24, 44 | ralrnmptw 7113 | . . . . . . . . . . . . . . . . . 18
⊢
(∀𝑦 ∈
𝑢 (◡𝐹 “ 𝑦) ∈ 𝐽 → (∀𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))(𝐹 “ 𝑐) ∈ 𝑢 ↔ ∀𝑦 ∈ 𝑢 (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝑢)) | 
| 46 | 20, 45 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (∀𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))(𝐹 “ 𝑐) ∈ 𝑢 ↔ ∀𝑦 ∈ 𝑢 (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝑢)) | 
| 47 | 42, 46 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))(𝐹 “ 𝑐) ∈ 𝑢) | 
| 48 | 47 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → ∀𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))(𝐹 “ 𝑐) ∈ 𝑢) | 
| 49 | 48 | r19.21bi 3250 | . . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ Comp
∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) ∧ 𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))) → (𝐹 “ 𝑐) ∈ 𝑢) | 
| 50 | 33, 49 | syldan 591 | . . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ Comp
∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) ∧ 𝑐 ∈ 𝑠) → (𝐹 “ 𝑐) ∈ 𝑢) | 
| 51 | 50 | fmpttd 7134 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)):𝑠⟶𝑢) | 
| 52 | 51 | frnd 6743 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ⊆ 𝑢) | 
| 53 |  | simprlr 779 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → 𝑠 ∈ Fin) | 
| 54 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢ (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) | 
| 55 | 54 | rnmpt 5967 | . . . . . . . . . . . . 13
⊢ ran
(𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)} | 
| 56 |  | abrexfi 9393 | . . . . . . . . . . . . 13
⊢ (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)} ∈ Fin) | 
| 57 | 55, 56 | eqeltrid 2844 | . . . . . . . . . . . 12
⊢ (𝑠 ∈ Fin → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ Fin) | 
| 58 | 53, 57 | syl 17 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ Fin) | 
| 59 |  | elfpw 9395 | . . . . . . . . . . 11
⊢ (ran
(𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ⊆ 𝑢 ∧ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ Fin)) | 
| 60 | 52, 58, 59 | sylanbrc 583 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ (𝒫 𝑢 ∩ Fin)) | 
| 61 | 17 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → 𝐹:∪ 𝐽⟶𝑌) | 
| 62 | 61 | fdmd 6745 | . . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → dom 𝐹 = ∪ 𝐽) | 
| 63 |  | simpll2 1213 | . . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → 𝐹:𝑋–onto→𝑌) | 
| 64 |  | fof 6819 | . . . . . . . . . . . . . 14
⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | 
| 65 |  | fdm 6744 | . . . . . . . . . . . . . 14
⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋) | 
| 66 | 63, 64, 65 | 3syl 18 | . . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → dom 𝐹 = 𝑋) | 
| 67 |  | simprr 772 | . . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → ∪ 𝐽 = ∪
𝑠) | 
| 68 | 62, 66, 67 | 3eqtr3d 2784 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → 𝑋 = ∪ 𝑠) | 
| 69 | 68 | imaeq2d 6077 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → (𝐹 “ 𝑋) = (𝐹 “ ∪ 𝑠)) | 
| 70 |  | foima 6824 | . . . . . . . . . . . 12
⊢ (𝐹:𝑋–onto→𝑌 → (𝐹 “ 𝑋) = 𝑌) | 
| 71 | 63, 70 | syl 17 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → (𝐹 “ 𝑋) = 𝑌) | 
| 72 | 50 | ralrimiva 3145 | . . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → ∀𝑐 ∈ 𝑠 (𝐹 “ 𝑐) ∈ 𝑢) | 
| 73 |  | dfiun2g 5029 | . . . . . . . . . . . . 13
⊢
(∀𝑐 ∈
𝑠 (𝐹 “ 𝑐) ∈ 𝑢 → ∪
𝑐 ∈ 𝑠 (𝐹 “ 𝑐) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}) | 
| 74 | 72, 73 | syl 17 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → ∪
𝑐 ∈ 𝑠 (𝐹 “ 𝑐) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}) | 
| 75 |  | imauni 7267 | . . . . . . . . . . . 12
⊢ (𝐹 “ ∪ 𝑠) =
∪ 𝑐 ∈ 𝑠 (𝐹 “ 𝑐) | 
| 76 | 55 | unieqi 4918 | . . . . . . . . . . . 12
⊢ ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)} | 
| 77 | 74, 75, 76 | 3eqtr4g 2801 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → (𝐹 “ ∪ 𝑠) = ∪
ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐))) | 
| 78 | 69, 71, 77 | 3eqtr3d 2784 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → 𝑌 = ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐))) | 
| 79 |  | unieq 4917 | . . . . . . . . . . 11
⊢ (𝑣 = ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) → ∪ 𝑣 = ∪
ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐))) | 
| 80 | 79 | rspceeqv 3644 | . . . . . . . . . 10
⊢ ((ran
(𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑌 = ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣) | 
| 81 | 60, 78, 80 | syl2anc 584 | . . . . . . . . 9
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 =
∪ 𝑠)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣) | 
| 82 | 81 | expr 456 | . . . . . . . 8
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ (𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin)) → (∪ 𝐽 =
∪ 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)) | 
| 83 | 31, 82 | sylan2b 594 | . . . . . . 7
⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∩ Fin)) → (∪ 𝐽 =
∪ 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)) | 
| 84 | 83 | rexlimdva 3154 | . . . . . 6
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (∃𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∩ Fin)∪
𝐽 = ∪ 𝑠
→ ∃𝑣 ∈
(𝒫 𝑢 ∩
Fin)𝑌 = ∪ 𝑣)) | 
| 85 | 30, 84 | mpd 15 | . . . . 5
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣) | 
| 86 | 85 | expr 456 | . . . 4
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ⊆ 𝐾) → (𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)) | 
| 87 | 3, 86 | sylan2 593 | . . 3
⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ∈ 𝒫 𝐾) → (𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)) | 
| 88 | 87 | ralrimiva 3145 | . 2
⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ 𝒫 𝐾(𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)) | 
| 89 | 15 | iscmp 23397 | . 2
⊢ (𝐾 ∈ Comp ↔ (𝐾 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐾(𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))) | 
| 90 | 2, 88, 89 | sylanbrc 583 | 1
⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp) |