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Theorem kgencn2 21283
Description: A function 𝐹:𝐽𝐾 from a compactly generated space is continuous iff for all compact spaces 𝑧 and continuous 𝑔:𝑧𝐽, the composite 𝐹𝑔:𝑧𝐾 is continuous. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
kgencn2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾))))
Distinct variable groups:   𝑧,𝑔,𝐹   𝑔,𝐽,𝑧   𝑔,𝐾,𝑧   𝑔,𝑋,𝑧   𝑔,𝑌,𝑧

Proof of Theorem kgencn2
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 kgencn 21282 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)))))
2 rncmp 21122 . . . . . . . 8 ((𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽)) → (𝐽t ran 𝑔) ∈ Comp)
32adantl 482 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (𝐽t ran 𝑔) ∈ Comp)
4 simprr 795 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn 𝐽))
5 eqid 2621 . . . . . . . . . . . 12 𝑧 = 𝑧
6 eqid 2621 . . . . . . . . . . . 12 𝐽 = 𝐽
75, 6cnf 20973 . . . . . . . . . . 11 (𝑔 ∈ (𝑧 Cn 𝐽) → 𝑔: 𝑧 𝐽)
8 frn 6015 . . . . . . . . . . 11 (𝑔: 𝑧 𝐽 → ran 𝑔 𝐽)
94, 7, 83syl 18 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 𝐽)
10 toponuni 20651 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1110ad3antrrr 765 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑋 = 𝐽)
129, 11sseqtr4d 3626 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔𝑋)
13 vex 3192 . . . . . . . . . . 11 𝑔 ∈ V
1413rnex 7054 . . . . . . . . . 10 ran 𝑔 ∈ V
1514elpw 4141 . . . . . . . . 9 (ran 𝑔 ∈ 𝒫 𝑋 ↔ ran 𝑔𝑋)
1612, 15sylibr 224 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ∈ 𝒫 𝑋)
17 oveq2 6618 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → (𝐽t 𝑘) = (𝐽t ran 𝑔))
1817eleq1d 2683 . . . . . . . . . 10 (𝑘 = ran 𝑔 → ((𝐽t 𝑘) ∈ Comp ↔ (𝐽t ran 𝑔) ∈ Comp))
19 reseq2 5356 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → (𝐹𝑘) = (𝐹 ↾ ran 𝑔))
2017oveq1d 6625 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → ((𝐽t 𝑘) Cn 𝐾) = ((𝐽t ran 𝑔) Cn 𝐾))
2119, 20eleq12d 2692 . . . . . . . . . 10 (𝑘 = ran 𝑔 → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾)))
2218, 21imbi12d 334 . . . . . . . . 9 (𝑘 = ran 𝑔 → (((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) ↔ ((𝐽t ran 𝑔) ∈ Comp → (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾))))
2322rspcv 3294 . . . . . . . 8 (ran 𝑔 ∈ 𝒫 𝑋 → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) → ((𝐽t ran 𝑔) ∈ Comp → (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾))))
2416, 23syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) → ((𝐽t ran 𝑔) ∈ Comp → (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾))))
253, 24mpid 44 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) → (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾)))
26 simplll 797 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝐽 ∈ (TopOn‘𝑋))
27 ssid 3608 . . . . . . . . . . 11 ran 𝑔 ⊆ ran 𝑔
2827a1i 11 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ ran 𝑔)
29 cnrest2 21013 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran 𝑔 ⊆ ran 𝑔 ∧ ran 𝑔𝑋) → (𝑔 ∈ (𝑧 Cn 𝐽) ↔ 𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔))))
3026, 28, 12, 29syl3anc 1323 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (𝑔 ∈ (𝑧 Cn 𝐽) ↔ 𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔))))
314, 30mpbid 222 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔)))
32 cnco 20993 . . . . . . . . 9 ((𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔)) ∧ (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾)) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
3332ex 450 . . . . . . . 8 (𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔)) → ((𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
3431, 33syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ((𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
35 cores 5602 . . . . . . . . 9 (ran 𝑔 ⊆ ran 𝑔 → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹𝑔))
3627, 35ax-mp 5 . . . . . . . 8 ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹𝑔)
3736eleq1i 2689 . . . . . . 7 (((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹𝑔) ∈ (𝑧 Cn 𝐾))
3834, 37syl6ib 241 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ((𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾) → (𝐹𝑔) ∈ (𝑧 Cn 𝐾)))
3925, 38syld 47 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) → (𝐹𝑔) ∈ (𝑧 Cn 𝐾)))
4039ralrimdvva 2969 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) → ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾)))
41 oveq1 6617 . . . . . . . . 9 (𝑧 = (𝐽t 𝑘) → (𝑧 Cn 𝐽) = ((𝐽t 𝑘) Cn 𝐽))
42 oveq1 6617 . . . . . . . . . 10 (𝑧 = (𝐽t 𝑘) → (𝑧 Cn 𝐾) = ((𝐽t 𝑘) Cn 𝐾))
4342eleq2d 2684 . . . . . . . . 9 (𝑧 = (𝐽t 𝑘) → ((𝐹𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾)))
4441, 43raleqbidv 3144 . . . . . . . 8 (𝑧 = (𝐽t 𝑘) → (∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾) ↔ ∀𝑔 ∈ ((𝐽t 𝑘) Cn 𝐽)(𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾)))
4544rspcv 3294 . . . . . . 7 ((𝐽t 𝑘) ∈ Comp → (∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾) → ∀𝑔 ∈ ((𝐽t 𝑘) Cn 𝐽)(𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾)))
46 elpwi 4145 . . . . . . . . . . . 12 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
4746adantl 482 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘𝑋)
4847resabs1d 5392 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) = ( I ↾ 𝑘))
49 idcn 20984 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
5049ad3antrrr 765 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
5110ad3antrrr 765 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑋 = 𝐽)
5247, 51sseqtrd 3625 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 𝐽)
536cnrest 21012 . . . . . . . . . . 11 ((( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ 𝑘 𝐽) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
5450, 52, 53syl2anc 692 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
5548, 54eqeltrrd 2699 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
56 coeq2 5245 . . . . . . . . . . 11 (𝑔 = ( I ↾ 𝑘) → (𝐹𝑔) = (𝐹 ∘ ( I ↾ 𝑘)))
5756eleq1d 2683 . . . . . . . . . 10 (𝑔 = ( I ↾ 𝑘) → ((𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽t 𝑘) Cn 𝐾)))
5857rspcv 3294 . . . . . . . . 9 (( I ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽) → (∀𝑔 ∈ ((𝐽t 𝑘) Cn 𝐽)(𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾) → (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽t 𝑘) Cn 𝐾)))
5955, 58syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (∀𝑔 ∈ ((𝐽t 𝑘) Cn 𝐽)(𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾) → (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽t 𝑘) Cn 𝐾)))
60 coires1 5617 . . . . . . . . 9 (𝐹 ∘ ( I ↾ 𝑘)) = (𝐹𝑘)
6160eleq1i 2689 . . . . . . . 8 ((𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))
6259, 61syl6ib 241 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (∀𝑔 ∈ ((𝐽t 𝑘) Cn 𝐽)(𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾) → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)))
6345, 62syl9r 78 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → (∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾) → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))))
6463com23 86 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾) → ((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))))
6564ralrimdva 2964 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))))
6640, 65impbid 202 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)) ↔ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾)))
6766pm5.32da 672 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾))))
681, 67bitrd 268 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3559  𝒫 cpw 4135   cuni 4407   I cid 4989  ran crn 5080  cres 5081  ccom 5083  wf 5848  cfv 5852  (class class class)co 6610  t crest 16013  TopOnctopon 20647   Cn ccn 20951  Compccmp 21112  𝑘Genckgen 21259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-fin 7911  df-fi 8269  df-rest 16015  df-topgen 16036  df-top 20631  df-topon 20648  df-bases 20674  df-cn 20954  df-cmp 21113  df-kgen 21260
This theorem is referenced by: (None)
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