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Theorem kgencn2 21640
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 21639 . 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 21479 . . . . . . . 8 ((𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽)) → (𝐽t ran 𝑔) ∈ Comp)
32adantl 473 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (𝐽t ran 𝑔) ∈ Comp)
4 simprr 789 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn 𝐽))
5 eqid 2765 . . . . . . . . . . . 12 𝑧 = 𝑧
6 eqid 2765 . . . . . . . . . . . 12 𝐽 = 𝐽
75, 6cnf 21330 . . . . . . . . . . 11 (𝑔 ∈ (𝑧 Cn 𝐽) → 𝑔: 𝑧 𝐽)
8 frn 6229 . . . . . . . . . . 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 20998 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1110ad3antrrr 721 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑋 = 𝐽)
129, 11sseqtr4d 3802 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔𝑋)
13 vex 3353 . . . . . . . . . . 11 𝑔 ∈ V
1413rnex 7298 . . . . . . . . . 10 ran 𝑔 ∈ V
1514elpw 4321 . . . . . . . . 9 (ran 𝑔 ∈ 𝒫 𝑋 ↔ ran 𝑔𝑋)
1612, 15sylibr 225 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ∈ 𝒫 𝑋)
17 oveq2 6850 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → (𝐽t 𝑘) = (𝐽t ran 𝑔))
1817eleq1d 2829 . . . . . . . . . 10 (𝑘 = ran 𝑔 → ((𝐽t 𝑘) ∈ Comp ↔ (𝐽t ran 𝑔) ∈ Comp))
19 reseq2 5560 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → (𝐹𝑘) = (𝐹 ↾ ran 𝑔))
2017oveq1d 6857 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → ((𝐽t 𝑘) Cn 𝐾) = ((𝐽t ran 𝑔) Cn 𝐾))
2119, 20eleq12d 2838 . . . . . . . . . 10 (𝑘 = ran 𝑔 → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾)))
2218, 21imbi12d 335 . . . . . . . . 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 3457 . . . . . . . 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 791 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝐽 ∈ (TopOn‘𝑋))
27 ssidd 3784 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ ran 𝑔)
28 cnrest2 21370 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran 𝑔 ⊆ ran 𝑔 ∧ ran 𝑔𝑋) → (𝑔 ∈ (𝑧 Cn 𝐽) ↔ 𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔))))
2926, 27, 12, 28syl3anc 1490 . . . . . . . . 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))))
304, 29mpbid 223 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔)))
31 cnco 21350 . . . . . . . . 9 ((𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔)) ∧ (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾)) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
3231ex 401 . . . . . . . 8 (𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔)) → ((𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
3330, 32syl 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)))
34 ssid 3783 . . . . . . . . 9 ran 𝑔 ⊆ ran 𝑔
35 cores 5824 . . . . . . . . 9 (ran 𝑔 ⊆ ran 𝑔 → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹𝑔))
3634, 35ax-mp 5 . . . . . . . 8 ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹𝑔)
3736eleq1i 2835 . . . . . . 7 (((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹𝑔) ∈ (𝑧 Cn 𝐾))
3833, 37syl6ib 242 . . . . . 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 3121 . . . 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 6849 . . . . . . . . 9 (𝑧 = (𝐽t 𝑘) → (𝑧 Cn 𝐽) = ((𝐽t 𝑘) Cn 𝐽))
42 oveq1 6849 . . . . . . . . . 10 (𝑧 = (𝐽t 𝑘) → (𝑧 Cn 𝐾) = ((𝐽t 𝑘) Cn 𝐾))
4342eleq2d 2830 . . . . . . . . 9 (𝑧 = (𝐽t 𝑘) → ((𝐹𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾)))
4441, 43raleqbidv 3300 . . . . . . . 8 (𝑧 = (𝐽t 𝑘) → (∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾) ↔ ∀𝑔 ∈ ((𝐽t 𝑘) Cn 𝐽)(𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾)))
4544rspcv 3457 . . . . . . 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 4325 . . . . . . . . . . . 12 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
4746adantl 473 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘𝑋)
4847resabs1d 5603 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) = ( I ↾ 𝑘))
49 idcn 21341 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
5049ad3antrrr 721 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
5110ad3antrrr 721 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑋 = 𝐽)
5247, 51sseqtrd 3801 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 𝐽)
536cnrest 21369 . . . . . . . . . . 11 ((( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ 𝑘 𝐽) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
5450, 52, 53syl2anc 579 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
5548, 54eqeltrrd 2845 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
56 coeq2 5449 . . . . . . . . . . 11 (𝑔 = ( I ↾ 𝑘) → (𝐹𝑔) = (𝐹 ∘ ( I ↾ 𝑘)))
5756eleq1d 2829 . . . . . . . . . 10 (𝑔 = ( I ↾ 𝑘) → ((𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽t 𝑘) Cn 𝐾)))
5857rspcv 3457 . . . . . . . . 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 5839 . . . . . . . . 9 (𝐹 ∘ ( I ↾ 𝑘)) = (𝐹𝑘)
6160eleq1i 2835 . . . . . . . 8 ((𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))
6259, 61syl6ib 242 . . . . . . 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 3116 . . . 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 203 . . 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 574 . 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 270 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wss 3732  𝒫 cpw 4315   cuni 4594   I cid 5184  ran crn 5278  cres 5279  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  t crest 16347  TopOnctopon 20994   Cn ccn 21308  Compccmp 21469  𝑘Genckgen 21616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-fin 8164  df-fi 8524  df-rest 16349  df-topgen 16370  df-top 20978  df-topon 20995  df-bases 21030  df-cn 21311  df-cmp 21470  df-kgen 21617
This theorem is referenced by: (None)
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