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Theorem xkoptsub 22259
Description: The compact-open topology is finer than the product topology restricted to continuous functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
xkoptsub.x 𝑋 = 𝑅
xkoptsub.j 𝐽 = (∏t‘(𝑋 × {𝑆}))
Assertion
Ref Expression
xkoptsub ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅))

Proof of Theorem xkoptsub
Dummy variables 𝑓 𝑔 𝑘 𝑛 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xkoptsub.j . . . . 5 𝐽 = (∏t‘(𝑋 × {𝑆}))
2 xkoptsub.x . . . . . . . . 9 𝑋 = 𝑅
32topopn 21511 . . . . . . . 8 (𝑅 ∈ Top → 𝑋𝑅)
43adantr 484 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑋𝑅)
5 fconstg 6540 . . . . . . . . 9 (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶{𝑆})
65adantl 485 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶{𝑆})
76ffnd 6488 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}) Fn 𝑋)
8 eqid 2798 . . . . . . . 8 {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦𝑋 (𝑔𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋𝑧)(𝑔𝑦) = ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦𝑋 (𝑔𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦𝑋 (𝑔𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋𝑧)(𝑔𝑦) = ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦𝑋 (𝑔𝑦))}
98ptval 22175 . . . . . . 7 ((𝑋𝑅 ∧ (𝑋 × {𝑆}) Fn 𝑋) → (∏t‘(𝑋 × {𝑆})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦𝑋 (𝑔𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋𝑧)(𝑔𝑦) = ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦𝑋 (𝑔𝑦))}))
104, 7, 9syl2anc 587 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∏t‘(𝑋 × {𝑆})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦𝑋 (𝑔𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋𝑧)(𝑔𝑦) = ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦𝑋 (𝑔𝑦))}))
11 simpr 488 . . . . . . . . . . 11 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
1211snssd 4702 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑆} ⊆ Top)
136, 12fssd 6502 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶Top)
14 eqid 2798 . . . . . . . . . 10 X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) = X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛)
158, 14ptbasfi 22186 . . . . . . . . 9 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦𝑋 (𝑔𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋𝑧)(𝑔𝑦) = ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦𝑋 (𝑔𝑦))} = (fi‘({X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛)} ∪ ran (𝑘𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ ((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢)))))
164, 13, 15syl2anc 587 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦𝑋 (𝑔𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋𝑧)(𝑔𝑦) = ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦𝑋 (𝑔𝑦))} = (fi‘({X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛)} ∪ ran (𝑘𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ ((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢)))))
17 fvconst2g 6941 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Top ∧ 𝑛𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆)
1817adantll 713 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆)
1918unieqd 4814 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆)
2019ixpeq2dva 8459 . . . . . . . . . . . 12 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) = X𝑛𝑋 𝑆)
21 eqid 2798 . . . . . . . . . . . . . 14 𝑆 = 𝑆
2221topopn 21511 . . . . . . . . . . . . 13 (𝑆 ∈ Top → 𝑆𝑆)
23 ixpconstg 8453 . . . . . . . . . . . . 13 ((𝑋𝑅 𝑆𝑆) → X𝑛𝑋 𝑆 = ( 𝑆m 𝑋))
243, 22, 23syl2an 598 . . . . . . . . . . . 12 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛𝑋 𝑆 = ( 𝑆m 𝑋))
2520, 24eqtrd 2833 . . . . . . . . . . 11 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) = ( 𝑆m 𝑋))
2625sneqd 4537 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛)} = {( 𝑆m 𝑋)})
27 eqid 2798 . . . . . . . . . . . 12 𝑋 = 𝑋
28 fvconst2g 6941 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Top ∧ 𝑘𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆)
2928adantll 713 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆)
3025adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘𝑋) → X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) = ( 𝑆m 𝑋))
3130mpteq1d 5119 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘𝑋) → (𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) = (𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)))
3231cnveqd 5710 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘𝑋) → (𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) = (𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)))
3332imaeq1d 5895 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘𝑋) → ((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))
3433ralrimivw 3150 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘𝑋) → ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))
3529, 34jca 515 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘𝑋) → (((𝑋 × {𝑆})‘𝑘) = 𝑆 ∧ ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))
3635ralrimiva 3149 . . . . . . . . . . . 12 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑘𝑋 (((𝑋 × {𝑆})‘𝑘) = 𝑆 ∧ ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))
37 mpoeq123 7205 . . . . . . . . . . . 12 ((𝑋 = 𝑋 ∧ ∀𝑘𝑋 (((𝑋 × {𝑆})‘𝑘) = 𝑆 ∧ ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) → (𝑘𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ ((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))
3827, 36, 37sylancr 590 . . . . . . . . . . 11 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑘𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ ((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))
3938rneqd 5772 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ ((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))
4026, 39uneq12d 4091 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ({X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛)} ∪ ran (𝑘𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ ((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢))) = ({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))))
4140fveq2d 6649 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (fi‘({X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛)} ∪ ran (𝑘𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ ((𝑤X𝑛𝑋 ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤𝑘)) “ 𝑢)))) = (fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))))
4216, 41eqtrd 2833 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦𝑋 (𝑔𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋𝑧)(𝑔𝑦) = ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦𝑋 (𝑔𝑦))} = (fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))))
4342fveq2d 6649 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦𝑋 (𝑔𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋𝑧)(𝑔𝑦) = ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦𝑋 (𝑔𝑦))}) = (topGen‘(fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))))))
4410, 43eqtrd 2833 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∏t‘(𝑋 × {𝑆})) = (topGen‘(fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))))))
451, 44syl5eq 2845 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 = (topGen‘(fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))))))
4645oveq1d 7150 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽t (𝑅 Cn 𝑆)) = ((topGen‘(fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆)))
47 firest 16698 . . . . 5 (fi‘(({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆))) = ((fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆))
4847fveq2i 6648 . . . 4 (topGen‘(fi‘(({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) = (topGen‘((fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆)))
49 fvex 6658 . . . . 5 (fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))) ∈ V
50 ovex 7168 . . . . 5 (𝑅 Cn 𝑆) ∈ V
51 tgrest 21764 . . . . 5 (((fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))) ∈ V ∧ (𝑅 Cn 𝑆) ∈ V) → (topGen‘((fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆))) = ((topGen‘(fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆)))
5249, 50, 51mp2an 691 . . . 4 (topGen‘((fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆))) = ((topGen‘(fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆))
5348, 52eqtri 2821 . . 3 (topGen‘(fi‘(({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) = ((topGen‘(fi‘({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆))
5446, 53eqtr4di 2851 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽t (𝑅 Cn 𝑆)) = (topGen‘(fi‘(({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))))
55 xkotop 22193 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ Top)
56 snex 5297 . . . . . 6 {( 𝑆m 𝑋)} ∈ V
57 mpoexga 7758 . . . . . . . 8 ((𝑋𝑅𝑆 ∈ Top) → (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) ∈ V)
583, 57sylan 583 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) ∈ V)
59 rnexg 7595 . . . . . . 7 ((𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) ∈ V → ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) ∈ V)
6058, 59syl 17 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) ∈ V)
61 unexg 7452 . . . . . 6 (({( 𝑆m 𝑋)} ∈ V ∧ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) ∈ V) → ({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
6256, 60, 61sylancr 590 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
63 restval 16692 . . . . 5 ((({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ∈ V ∧ (𝑅 Cn 𝑆) ∈ V) → (({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) = ran (𝑥 ∈ ({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))))
6462, 50, 63sylancl 589 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) = ran (𝑥 ∈ ({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))))
65 elun 4076 . . . . . . 7 (𝑥 ∈ ({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↔ (𝑥 ∈ {( 𝑆m 𝑋)} ∨ 𝑥 ∈ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))))
662, 21cnf 21851 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥:𝑋 𝑆)
67 elmapg 8402 . . . . . . . . . . . . . . 15 (( 𝑆𝑆𝑋𝑅) → (𝑥 ∈ ( 𝑆m 𝑋) ↔ 𝑥:𝑋 𝑆))
6822, 3, 67syl2anr 599 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ ( 𝑆m 𝑋) ↔ 𝑥:𝑋 𝑆))
6966, 68syl5ibr 249 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥 ∈ ( 𝑆m 𝑋)))
7069ssrdv 3921 . . . . . . . . . . . 12 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ ( 𝑆m 𝑋))
7170adantr 484 . . . . . . . . . . 11 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {( 𝑆m 𝑋)}) → (𝑅 Cn 𝑆) ⊆ ( 𝑆m 𝑋))
72 elsni 4542 . . . . . . . . . . . 12 (𝑥 ∈ {( 𝑆m 𝑋)} → 𝑥 = ( 𝑆m 𝑋))
7372adantl 485 . . . . . . . . . . 11 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {( 𝑆m 𝑋)}) → 𝑥 = ( 𝑆m 𝑋))
7471, 73sseqtrrd 3956 . . . . . . . . . 10 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {( 𝑆m 𝑋)}) → (𝑅 Cn 𝑆) ⊆ 𝑥)
75 sseqin2 4142 . . . . . . . . . 10 ((𝑅 Cn 𝑆) ⊆ 𝑥 ↔ (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆))
7674, 75sylib 221 . . . . . . . . 9 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {( 𝑆m 𝑋)}) → (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆))
77 eqid 2798 . . . . . . . . . . . 12 (𝑆ko 𝑅) = (𝑆ko 𝑅)
7877xkouni 22204 . . . . . . . . . . 11 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = (𝑆ko 𝑅))
79 eqid 2798 . . . . . . . . . . . . 13 (𝑆ko 𝑅) = (𝑆ko 𝑅)
8079topopn 21511 . . . . . . . . . . . 12 ((𝑆ko 𝑅) ∈ Top → (𝑆ko 𝑅) ∈ (𝑆ko 𝑅))
8155, 80syl 17 . . . . . . . . . . 11 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (𝑆ko 𝑅))
8278, 81eqeltrd 2890 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆ko 𝑅))
8382adantr 484 . . . . . . . . 9 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {( 𝑆m 𝑋)}) → (𝑅 Cn 𝑆) ∈ (𝑆ko 𝑅))
8476, 83eqeltrd 2890 . . . . . . . 8 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {( 𝑆m 𝑋)}) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅))
85 eqid 2798 . . . . . . . . . . 11 (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) = (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))
8685rnmpo 7263 . . . . . . . . . 10 ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) = {𝑥 ∣ ∃𝑘𝑋𝑢𝑆 𝑥 = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)}
8786abeq2i 2925 . . . . . . . . 9 (𝑥 ∈ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) ↔ ∃𝑘𝑋𝑢𝑆 𝑥 = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))
88 cnvresima 6054 . . . . . . . . . . . . . . 15 (((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) ↾ (𝑅 Cn 𝑆)) “ 𝑢) = (((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆))
8970adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (𝑅 Cn 𝑆) ⊆ ( 𝑆m 𝑋))
9089resmptd 5875 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) ↾ (𝑅 Cn 𝑆)) = (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)))
9190cnveqd 5710 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) ↾ (𝑅 Cn 𝑆)) = (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)))
9291imaeq1d 5895 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) ↾ (𝑅 Cn 𝑆)) “ 𝑢) = ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) “ 𝑢))
9388, 92syl5eqr 2847 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) = ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) “ 𝑢))
94 fvex 6658 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝑘) ∈ V
9594rgenw 3118 . . . . . . . . . . . . . . . . . . 19 𝑤 ∈ (𝑅 Cn 𝑆)(𝑤𝑘) ∈ V
96 eqid 2798 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) = (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘))
9796fnmpt 6460 . . . . . . . . . . . . . . . . . . 19 (∀𝑤 ∈ (𝑅 Cn 𝑆)(𝑤𝑘) ∈ V → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) Fn (𝑅 Cn 𝑆))
9895, 97mp1i 13 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) Fn (𝑅 Cn 𝑆))
99 elpreima 6805 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) Fn (𝑅 Cn 𝑆) → (𝑓 ∈ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘))‘𝑓) ∈ 𝑢)))
10098, 99syl 17 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (𝑓 ∈ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘))‘𝑓) ∈ 𝑢)))
101 fveq1 6644 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑓 → (𝑤𝑘) = (𝑓𝑘))
102 fvex 6658 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑘) ∈ V
103101, 96, 102fvmpt 6745 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ (𝑅 Cn 𝑆) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘))‘𝑓) = (𝑓𝑘))
104103adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘))‘𝑓) = (𝑓𝑘))
105104eleq1d 2874 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘))‘𝑓) ∈ 𝑢 ↔ (𝑓𝑘) ∈ 𝑢))
106102snss 4679 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑘) ∈ 𝑢 ↔ {(𝑓𝑘)} ⊆ 𝑢)
10789sselda 3915 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 ∈ ( 𝑆m 𝑋))
108 elmapi 8411 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 ∈ ( 𝑆m 𝑋) → 𝑓:𝑋 𝑆)
109 ffn 6487 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓:𝑋 𝑆𝑓 Fn 𝑋)
110107, 108, 1093syl 18 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 Fn 𝑋)
111 simplrl 776 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑘𝑋)
112 fnsnfv 6718 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓 Fn 𝑋𝑘𝑋) → {(𝑓𝑘)} = (𝑓 “ {𝑘}))
113110, 111, 112syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → {(𝑓𝑘)} = (𝑓 “ {𝑘}))
114113sseq1d 3946 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ({(𝑓𝑘)} ⊆ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
115106, 114syl5bb 286 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ((𝑓𝑘) ∈ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
116105, 115bitrd 282 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘))‘𝑓) ∈ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
117116pm5.32da 582 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → ((𝑓 ∈ (𝑅 Cn 𝑆) ∧ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘))‘𝑓) ∈ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)))
118100, 117bitrd 282 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (𝑓 ∈ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)))
119118abbi2dv 2927 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) “ 𝑢) = {𝑓 ∣ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)})
120 df-rab 3115 . . . . . . . . . . . . . . 15 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} = {𝑓 ∣ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)}
121119, 120eqtr4di 2851 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤𝑘)) “ 𝑢) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢})
12293, 121eqtrd 2833 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢})
123 simpll 766 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → 𝑅 ∈ Top)
12411adantr 484 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → 𝑆 ∈ Top)
125 simprl 770 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → 𝑘𝑋)
126125snssd 4702 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → {𝑘} ⊆ 𝑋)
1272toptopon 21522 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
128123, 127sylib 221 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → 𝑅 ∈ (TopOn‘𝑋))
129 restsn2 21776 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝑅t {𝑘}) = 𝒫 {𝑘})
130128, 125, 129syl2anc 587 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (𝑅t {𝑘}) = 𝒫 {𝑘})
131 snfi 8577 . . . . . . . . . . . . . . . 16 {𝑘} ∈ Fin
132 discmp 22003 . . . . . . . . . . . . . . . 16 ({𝑘} ∈ Fin ↔ 𝒫 {𝑘} ∈ Comp)
133131, 132mpbi 233 . . . . . . . . . . . . . . 15 𝒫 {𝑘} ∈ Comp
134130, 133eqeltrdi 2898 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (𝑅t {𝑘}) ∈ Comp)
135 simprr 772 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → 𝑢𝑆)
1362, 123, 124, 126, 134, 135xkoopn 22194 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} ∈ (𝑆ko 𝑅))
137122, 136eqeltrd 2890 . . . . . . . . . . . 12 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅))
138 ineq1 4131 . . . . . . . . . . . . 13 (𝑥 = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) = (((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)))
139138eleq1d 2874 . . . . . . . . . . . 12 (𝑥 = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) → ((𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅) ↔ (((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅)))
140137, 139syl5ibrcom 250 . . . . . . . . . . 11 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘𝑋𝑢𝑆)) → (𝑥 = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅)))
141140rexlimdvva 3253 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∃𝑘𝑋𝑢𝑆 𝑥 = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅)))
142141imp 410 . . . . . . . . 9 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ∃𝑘𝑋𝑢𝑆 𝑥 = ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅))
14387, 142sylan2b 596 . . . . . . . 8 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅))
14484, 143jaodan 955 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑥 ∈ {( 𝑆m 𝑋)} ∨ 𝑥 ∈ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅))
14565, 144sylan2b 596 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ ({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆ko 𝑅))
146145fmpttd 6856 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ ({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))):({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢)))⟶(𝑆ko 𝑅))
147146frnd 6494 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑥 ∈ ({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))) ⊆ (𝑆ko 𝑅))
14864, 147eqsstrd 3953 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅))
149 tgfiss 21596 . . 3 (((𝑆ko 𝑅) ∈ Top ∧ (({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅)) → (topGen‘(fi‘(({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) ⊆ (𝑆ko 𝑅))
15055, 148, 149syl2anc 587 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (topGen‘(fi‘(({( 𝑆m 𝑋)} ∪ ran (𝑘𝑋, 𝑢𝑆 ↦ ((𝑤 ∈ ( 𝑆m 𝑋) ↦ (𝑤𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) ⊆ (𝑆ko 𝑅))
15154, 150eqsstrd 3953 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wex 1781  wcel 2111  {cab 2776  wral 3106  wrex 3107  {crab 3110  Vcvv 3441  cdif 3878  cun 3879  cin 3880  wss 3881  𝒫 cpw 4497  {csn 4525   cuni 4800  cmpt 5110   × cxp 5517  ccnv 5518  ran crn 5520  cres 5521  cima 5522   Fn wfn 6319  wf 6320  cfv 6324  (class class class)co 7135  cmpo 7137  m cmap 8389  Xcixp 8444  Fincfn 8492  ficfi 8858  t crest 16686  topGenctg 16703  tcpt 16704  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  Compccmp 21991  ko cxko 22166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-pt 16710  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-cmp 21992  df-xko 22168
This theorem is referenced by:  xkopt  22260  xkopjcn  22261
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