| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | xkoptsub.j | . . . . 5
⊢ 𝐽 =
(∏t‘(𝑋 × {𝑆})) | 
| 2 |  | xkoptsub.x | . . . . . . . . 9
⊢ 𝑋 = ∪
𝑅 | 
| 3 | 2 | topopn 22912 | . . . . . . . 8
⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅) | 
| 4 | 3 | adantr 480 | . . . . . . 7
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑋 ∈ 𝑅) | 
| 5 |  | fconstg 6795 | . . . . . . . . 9
⊢ (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶{𝑆}) | 
| 6 | 5 | adantl 481 | . . . . . . . 8
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶{𝑆}) | 
| 7 | 6 | ffnd 6737 | . . . . . . 7
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}) Fn 𝑋) | 
| 8 |  | eqid 2737 | . . . . . . . 8
⊢ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))} | 
| 9 | 8 | ptval 23578 | . . . . . . 7
⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}) Fn 𝑋) → (∏t‘(𝑋 × {𝑆})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))})) | 
| 10 | 4, 7, 9 | syl2anc 584 | . . . . . 6
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) →
(∏t‘(𝑋 × {𝑆})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))})) | 
| 11 |  | simpr 484 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top) | 
| 12 | 11 | snssd 4809 | . . . . . . . . . 10
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑆} ⊆ Top) | 
| 13 | 6, 12 | fssd 6753 | . . . . . . . . 9
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶Top) | 
| 14 |  | eqid 2737 | . . . . . . . . . 10
⊢ X𝑛 ∈
𝑋 ∪ ((𝑋
× {𝑆})‘𝑛) = X𝑛 ∈
𝑋 ∪ ((𝑋
× {𝑆})‘𝑛) | 
| 15 | 8, 14 | ptbasfi 23589 | . . . . . . . . 9
⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))} = (fi‘({X𝑛 ∈
𝑋 ∪ ((𝑋
× {𝑆})‘𝑛)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))))) | 
| 16 | 4, 13, 15 | syl2anc 584 | . . . . . . . 8
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))} = (fi‘({X𝑛 ∈
𝑋 ∪ ((𝑋
× {𝑆})‘𝑛)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))))) | 
| 17 |  | fvconst2g 7222 | . . . . . . . . . . . . . . 15
⊢ ((𝑆 ∈ Top ∧ 𝑛 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆) | 
| 18 | 17 | adantll 714 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆) | 
| 19 | 18 | unieqd 4920 | . . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛 ∈ 𝑋) → ∪
((𝑋 × {𝑆})‘𝑛) = ∪ 𝑆) | 
| 20 | 19 | ixpeq2dva 8952 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈
𝑋 ∪ ((𝑋
× {𝑆})‘𝑛) = X𝑛 ∈
𝑋 ∪ 𝑆) | 
| 21 |  | eqid 2737 | . . . . . . . . . . . . . 14
⊢ ∪ 𝑆 =
∪ 𝑆 | 
| 22 | 21 | topopn 22912 | . . . . . . . . . . . . 13
⊢ (𝑆 ∈ Top → ∪ 𝑆
∈ 𝑆) | 
| 23 |  | ixpconstg 8946 | . . . . . . . . . . . . 13
⊢ ((𝑋 ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆) → X𝑛 ∈ 𝑋 ∪ 𝑆 = (∪
𝑆 ↑m 𝑋)) | 
| 24 | 3, 22, 23 | syl2an 596 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈
𝑋 ∪ 𝑆 =
(∪ 𝑆 ↑m 𝑋)) | 
| 25 | 20, 24 | eqtrd 2777 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈
𝑋 ∪ ((𝑋
× {𝑆})‘𝑛) = (∪ 𝑆
↑m 𝑋)) | 
| 26 | 25 | sneqd 4638 | . . . . . . . . . 10
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {X𝑛 ∈
𝑋 ∪ ((𝑋
× {𝑆})‘𝑛)} = {(∪ 𝑆
↑m 𝑋)}) | 
| 27 |  | eqid 2737 | . . . . . . . . . . . 12
⊢ 𝑋 = 𝑋 | 
| 28 |  | fvconst2g 7222 | . . . . . . . . . . . . . . 15
⊢ ((𝑆 ∈ Top ∧ 𝑘 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆) | 
| 29 | 28 | adantll 714 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆) | 
| 30 | 25 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = (∪ 𝑆 ↑m 𝑋)) | 
| 31 | 30 | mpteq1d 5237 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → (𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) = (𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘))) | 
| 32 | 31 | cnveqd 5886 | . . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘))) | 
| 33 | 32 | imaeq1d 6077 | . . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) | 
| 34 | 33 | ralrimivw 3150 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) | 
| 35 | 29, 34 | jca 511 | . . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → (((𝑋 × {𝑆})‘𝑘) = 𝑆 ∧ ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) | 
| 36 | 35 | ralrimiva 3146 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) →
∀𝑘 ∈ 𝑋 (((𝑋 × {𝑆})‘𝑘) = 𝑆 ∧ ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) | 
| 37 |  | mpoeq123 7505 | . . . . . . . . . . . 12
⊢ ((𝑋 = 𝑋 ∧ ∀𝑘 ∈ 𝑋 (((𝑋 × {𝑆})‘𝑘) = 𝑆 ∧ ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) → (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) | 
| 38 | 27, 36, 37 | sylancr 587 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) | 
| 39 | 38 | rneqd 5949 | . . . . . . . . . 10
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) | 
| 40 | 26, 39 | uneq12d 4169 | . . . . . . . . 9
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ({X𝑛 ∈
𝑋 ∪ ((𝑋
× {𝑆})‘𝑛)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))) = ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) | 
| 41 | 40 | fveq2d 6910 | . . . . . . . 8
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) →
(fi‘({X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)))) = (fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) | 
| 42 | 16, 41 | eqtrd 2777 | . . . . . . 7
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))} = (fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) | 
| 43 | 42 | fveq2d 6910 | . . . . . 6
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) →
(topGen‘{𝑥 ∣
∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))}) = (topGen‘(fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))))) | 
| 44 | 10, 43 | eqtrd 2777 | . . . . 5
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) →
(∏t‘(𝑋 × {𝑆})) = (topGen‘(fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))))) | 
| 45 | 1, 44 | eqtrid 2789 | . . . 4
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 =
(topGen‘(fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))))) | 
| 46 | 45 | oveq1d 7446 | . . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽 ↾t (𝑅 Cn 𝑆)) = ((topGen‘(fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆))) | 
| 47 |  | firest 17477 | . . . . 5
⊢
(fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆))) = ((fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆)) | 
| 48 | 47 | fveq2i 6909 | . . . 4
⊢
(topGen‘(fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) = (topGen‘((fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆))) | 
| 49 |  | fvex 6919 | . . . . 5
⊢
(fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ∈ V | 
| 50 |  | ovex 7464 | . . . . 5
⊢ (𝑅 Cn 𝑆) ∈ V | 
| 51 |  | tgrest 23167 | . . . . 5
⊢
(((fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ∈ V ∧ (𝑅 Cn 𝑆) ∈ V) →
(topGen‘((fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆))) = ((topGen‘(fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆))) | 
| 52 | 49, 50, 51 | mp2an 692 | . . . 4
⊢
(topGen‘((fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆))) = ((topGen‘(fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆)) | 
| 53 | 48, 52 | eqtri 2765 | . . 3
⊢
(topGen‘(fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) = ((topGen‘(fi‘({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆)) | 
| 54 | 46, 53 | eqtr4di 2795 | . 2
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽 ↾t (𝑅 Cn 𝑆)) = (topGen‘(fi‘(({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆))))) | 
| 55 |  | xkotop 23596 | . . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ Top) | 
| 56 |  | snex 5436 | . . . . . 6
⊢ {(∪ 𝑆
↑m 𝑋)}
∈ V | 
| 57 |  | mpoexga 8102 | . . . . . . . 8
⊢ ((𝑋 ∈ 𝑅 ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V) | 
| 58 | 3, 57 | sylan 580 | . . . . . . 7
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V) | 
| 59 |  | rnexg 7924 | . . . . . . 7
⊢ ((𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V → ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V) | 
| 60 | 58, 59 | syl 17 | . . . . . 6
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V) | 
| 61 |  | unexg 7763 | . . . . . 6
⊢ (({(∪ 𝑆
↑m 𝑋)}
∈ V ∧ ran (𝑘
∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V) → ({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V) | 
| 62 | 56, 60, 61 | sylancr 587 | . . . . 5
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V) | 
| 63 |  | restval 17471 | . . . . 5
⊢
((({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V ∧ (𝑅 Cn 𝑆) ∈ V) → (({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) = ran (𝑥 ∈ ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆)))) | 
| 64 | 62, 50, 63 | sylancl 586 | . . . 4
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) = ran (𝑥 ∈ ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆)))) | 
| 65 |  | elun 4153 | . . . . . . 7
⊢ (𝑥 ∈ ({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↔ (𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)} ∨ 𝑥 ∈ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) | 
| 66 | 2, 21 | cnf 23254 | . . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥:𝑋⟶∪ 𝑆) | 
| 67 |  | elmapg 8879 | . . . . . . . . . . . . . . 15
⊢ ((∪ 𝑆
∈ 𝑆 ∧ 𝑋 ∈ 𝑅) → (𝑥 ∈ (∪ 𝑆 ↑m 𝑋) ↔ 𝑥:𝑋⟶∪ 𝑆)) | 
| 68 | 22, 3, 67 | syl2anr 597 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ (∪ 𝑆
↑m 𝑋)
↔ 𝑥:𝑋⟶∪ 𝑆)) | 
| 69 | 66, 68 | imbitrrid 246 | . . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥 ∈ (∪ 𝑆 ↑m 𝑋))) | 
| 70 | 69 | ssrdv 3989 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ (∪
𝑆 ↑m 𝑋)) | 
| 71 | 70 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆
↑m 𝑋)})
→ (𝑅 Cn 𝑆) ⊆ (∪ 𝑆
↑m 𝑋)) | 
| 72 |  | elsni 4643 | . . . . . . . . . . . 12
⊢ (𝑥 ∈ {(∪ 𝑆
↑m 𝑋)}
→ 𝑥 = (∪ 𝑆
↑m 𝑋)) | 
| 73 | 72 | adantl 481 | . . . . . . . . . . 11
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆
↑m 𝑋)})
→ 𝑥 = (∪ 𝑆
↑m 𝑋)) | 
| 74 | 71, 73 | sseqtrrd 4021 | . . . . . . . . . 10
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆
↑m 𝑋)})
→ (𝑅 Cn 𝑆) ⊆ 𝑥) | 
| 75 |  | sseqin2 4223 | . . . . . . . . . 10
⊢ ((𝑅 Cn 𝑆) ⊆ 𝑥 ↔ (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆)) | 
| 76 | 74, 75 | sylib 218 | . . . . . . . . 9
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆
↑m 𝑋)})
→ (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆)) | 
| 77 |  | eqid 2737 | . . . . . . . . . . . 12
⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅) | 
| 78 | 77 | xkouni 23607 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ (𝑆 ↑ko 𝑅)) | 
| 79 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢ ∪ (𝑆
↑ko 𝑅) =
∪ (𝑆 ↑ko 𝑅) | 
| 80 | 79 | topopn 22912 | . . . . . . . . . . . 12
⊢ ((𝑆 ↑ko 𝑅) ∈ Top → ∪ (𝑆
↑ko 𝑅)
∈ (𝑆
↑ko 𝑅)) | 
| 81 | 55, 80 | syl 17 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ (𝑆
↑ko 𝑅)
∈ (𝑆
↑ko 𝑅)) | 
| 82 | 78, 81 | eqeltrd 2841 | . . . . . . . . . 10
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅)) | 
| 83 | 82 | adantr 480 | . . . . . . . . 9
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆
↑m 𝑋)})
→ (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅)) | 
| 84 | 76, 83 | eqeltrd 2841 | . . . . . . . 8
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆
↑m 𝑋)})
→ (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅)) | 
| 85 |  | eqid 2737 | . . . . . . . . . . 11
⊢ (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) | 
| 86 | 85 | rnmpo 7566 | . . . . . . . . . 10
⊢ ran
(𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) = {𝑥 ∣ ∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)} | 
| 87 | 86 | eqabri 2885 | . . . . . . . . 9
⊢ (𝑥 ∈ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ↔ ∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) | 
| 88 |  | cnvresima 6250 | . . . . . . . . . . . . . . 15
⊢ (◡((𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) “ 𝑢) = ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) | 
| 89 | 70 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 Cn 𝑆) ⊆ (∪
𝑆 ↑m 𝑋)) | 
| 90 | 89 | resmptd 6058 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) = (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))) | 
| 91 | 90 | cnveqd 5886 | . . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ◡((𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) = ◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))) | 
| 92 | 91 | imaeq1d 6077 | . . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (◡((𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) “ 𝑢) = (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢)) | 
| 93 | 88, 92 | eqtr3id 2791 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) = (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢)) | 
| 94 |  | fvex 6919 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤‘𝑘) ∈ V | 
| 95 | 94 | rgenw 3065 | . . . . . . . . . . . . . . . . . . 19
⊢
∀𝑤 ∈
(𝑅 Cn 𝑆)(𝑤‘𝑘) ∈ V | 
| 96 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) = (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) | 
| 97 | 96 | fnmpt 6708 | . . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑤 ∈
(𝑅 Cn 𝑆)(𝑤‘𝑘) ∈ V → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆)) | 
| 98 | 95, 97 | mp1i 13 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆)) | 
| 99 |  | elpreima 7078 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆) → (𝑓 ∈ (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢))) | 
| 100 | 98, 99 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑓 ∈ (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢))) | 
| 101 |  | fveq1 6905 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 = 𝑓 → (𝑤‘𝑘) = (𝑓‘𝑘)) | 
| 102 |  | fvex 6919 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓‘𝑘) ∈ V | 
| 103 | 101, 96, 102 | fvmpt 7016 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) = (𝑓‘𝑘)) | 
| 104 | 103 | adantl 481 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) = (𝑓‘𝑘)) | 
| 105 | 104 | eleq1d 2826 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢 ↔ (𝑓‘𝑘) ∈ 𝑢)) | 
| 106 | 102 | snss 4785 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓‘𝑘) ∈ 𝑢 ↔ {(𝑓‘𝑘)} ⊆ 𝑢) | 
| 107 | 89 | sselda 3983 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 ∈ (∪ 𝑆 ↑m 𝑋)) | 
| 108 |  | elmapi 8889 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑓 ∈ (∪ 𝑆
↑m 𝑋)
→ 𝑓:𝑋⟶∪ 𝑆) | 
| 109 |  | ffn 6736 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑓:𝑋⟶∪ 𝑆 → 𝑓 Fn 𝑋) | 
| 110 | 107, 108,
109 | 3syl 18 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 Fn 𝑋) | 
| 111 |  | simplrl 777 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑘 ∈ 𝑋) | 
| 112 |  | fnsnfv 6988 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓 Fn 𝑋 ∧ 𝑘 ∈ 𝑋) → {(𝑓‘𝑘)} = (𝑓 “ {𝑘})) | 
| 113 | 110, 111,
112 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → {(𝑓‘𝑘)} = (𝑓 “ {𝑘})) | 
| 114 | 113 | sseq1d 4015 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ({(𝑓‘𝑘)} ⊆ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢)) | 
| 115 | 106, 114 | bitrid 283 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ((𝑓‘𝑘) ∈ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢)) | 
| 116 | 105, 115 | bitrd 279 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢)) | 
| 117 | 116 | pm5.32da 579 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((𝑓 ∈ (𝑅 Cn 𝑆) ∧ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢))) | 
| 118 | 100, 117 | bitrd 279 | . . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑓 ∈ (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢))) | 
| 119 | 118 | eqabdv 2875 | . . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑓 ∣ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)}) | 
| 120 |  | df-rab 3437 | . . . . . . . . . . . . . . 15
⊢ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} = {𝑓 ∣ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)} | 
| 121 | 119, 120 | eqtr4di 2795 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢}) | 
| 122 | 93, 121 | eqtrd 2777 | . . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢}) | 
| 123 |  | simpll 767 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑅 ∈ Top) | 
| 124 | 11 | adantr 480 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ∈ Top) | 
| 125 |  | simprl 771 | . . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑘 ∈ 𝑋) | 
| 126 | 125 | snssd 4809 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → {𝑘} ⊆ 𝑋) | 
| 127 | 2 | toptopon 22923 | . . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋)) | 
| 128 | 123, 127 | sylib 218 | . . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑅 ∈ (TopOn‘𝑋)) | 
| 129 |  | restsn2 23179 | . . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝑋) → (𝑅 ↾t {𝑘}) = 𝒫 {𝑘}) | 
| 130 | 128, 125,
129 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 ↾t {𝑘}) = 𝒫 {𝑘}) | 
| 131 |  | snfi 9083 | . . . . . . . . . . . . . . . 16
⊢ {𝑘} ∈ Fin | 
| 132 |  | discmp 23406 | . . . . . . . . . . . . . . . 16
⊢ ({𝑘} ∈ Fin ↔ 𝒫
{𝑘} ∈
Comp) | 
| 133 | 131, 132 | mpbi 230 | . . . . . . . . . . . . . . 15
⊢ 𝒫
{𝑘} ∈
Comp | 
| 134 | 130, 133 | eqeltrdi 2849 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 ↾t {𝑘}) ∈ Comp) | 
| 135 |  | simprr 773 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆) | 
| 136 | 2, 123, 124, 126, 134, 135 | xkoopn 23597 | . . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} ∈ (𝑆 ↑ko 𝑅)) | 
| 137 | 122, 136 | eqeltrd 2841 | . . . . . . . . . . . 12
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅)) | 
| 138 |  | ineq1 4213 | . . . . . . . . . . . . 13
⊢ (𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) = ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆))) | 
| 139 | 138 | eleq1d 2826 | . . . . . . . . . . . 12
⊢ (𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → ((𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅) ↔ ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))) | 
| 140 | 137, 139 | syl5ibrcom 247 | . . . . . . . . . . 11
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))) | 
| 141 | 140 | rexlimdvva 3213 | . . . . . . . . . 10
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) →
(∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))) | 
| 142 | 141 | imp 406 | . . . . . . . . 9
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅)) | 
| 143 | 87, 142 | sylan2b 594 | . . . . . . . 8
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅)) | 
| 144 | 84, 143 | jaodan 960 | . . . . . . 7
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑥 ∈ {(∪ 𝑆
↑m 𝑋)} ∨
𝑥 ∈ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅)) | 
| 145 | 65, 144 | sylan2b 594 | . . . . . 6
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ ({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅)) | 
| 146 | 145 | fmpttd 7135 | . . . . 5
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ ({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))):({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))⟶(𝑆 ↑ko 𝑅)) | 
| 147 | 146 | frnd 6744 | . . . 4
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑥 ∈ ({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))) ⊆ (𝑆 ↑ko 𝑅)) | 
| 148 | 64, 147 | eqsstrd 4018 | . . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅)) | 
| 149 |  | tgfiss 22998 | . . 3
⊢ (((𝑆 ↑ko 𝑅) ∈ Top ∧ (({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅)) → (topGen‘(fi‘(({(∪ 𝑆
↑m 𝑋)}
∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) ⊆ (𝑆 ↑ko 𝑅)) | 
| 150 | 55, 148, 149 | syl2anc 584 | . 2
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) →
(topGen‘(fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) ⊆ (𝑆 ↑ko 𝑅)) | 
| 151 | 54, 150 | eqsstrd 4018 | 1
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽 ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅)) |