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Theorem kgencn2 22093
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 22092 . 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 21932 . . . . . . . 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 769 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn 𝐽))
5 eqid 2818 . . . . . . . . . . . 12 𝑧 = 𝑧
6 eqid 2818 . . . . . . . . . . . 12 𝐽 = 𝐽
75, 6cnf 21782 . . . . . . . . . . 11 (𝑔 ∈ (𝑧 Cn 𝐽) → 𝑔: 𝑧 𝐽)
8 frn 6513 . . . . . . . . . . 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 21450 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1110ad3antrrr 726 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑋 = 𝐽)
129, 11sseqtrrd 4005 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔𝑋)
13 vex 3495 . . . . . . . . . . 11 𝑔 ∈ V
1413rnex 7606 . . . . . . . . . 10 ran 𝑔 ∈ V
1514elpw 4542 . . . . . . . . 9 (ran 𝑔 ∈ 𝒫 𝑋 ↔ ran 𝑔𝑋)
1612, 15sylibr 235 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ∈ 𝒫 𝑋)
17 oveq2 7153 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → (𝐽t 𝑘) = (𝐽t ran 𝑔))
1817eleq1d 2894 . . . . . . . . . 10 (𝑘 = ran 𝑔 → ((𝐽t 𝑘) ∈ Comp ↔ (𝐽t ran 𝑔) ∈ Comp))
19 reseq2 5841 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → (𝐹𝑘) = (𝐹 ↾ ran 𝑔))
2017oveq1d 7160 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → ((𝐽t 𝑘) Cn 𝐾) = ((𝐽t ran 𝑔) Cn 𝐾))
2119, 20eleq12d 2904 . . . . . . . . . 10 (𝑘 = ran 𝑔 → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾)))
2218, 21imbi12d 346 . . . . . . . . 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 3615 . . . . . . . 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 771 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝐽 ∈ (TopOn‘𝑋))
27 ssidd 3987 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔 ⊆ ran 𝑔)
28 cnrest2 21822 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran 𝑔 ⊆ ran 𝑔 ∧ ran 𝑔𝑋) → (𝑔 ∈ (𝑧 Cn 𝐽) ↔ 𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔))))
2926, 27, 12, 28syl3anc 1363 . . . . . . . . 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 233 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔)))
31 cnco 21802 . . . . . . . . 9 ((𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔)) ∧ (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾)) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
3231ex 413 . . . . . . . 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 3986 . . . . . . . . 9 ran 𝑔 ⊆ ran 𝑔
35 cores 6095 . . . . . . . . 9 (ran 𝑔 ⊆ ran 𝑔 → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹𝑔))
3634, 35ax-mp 5 . . . . . . . 8 ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹𝑔)
3736eleq1i 2900 . . . . . . 7 (((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹𝑔) ∈ (𝑧 Cn 𝐾))
3833, 37syl6ib 252 . . . . . 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 3191 . . . 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 7152 . . . . . . . . 9 (𝑧 = (𝐽t 𝑘) → (𝑧 Cn 𝐽) = ((𝐽t 𝑘) Cn 𝐽))
42 oveq1 7152 . . . . . . . . . 10 (𝑧 = (𝐽t 𝑘) → (𝑧 Cn 𝐾) = ((𝐽t 𝑘) Cn 𝐾))
4342eleq2d 2895 . . . . . . . . 9 (𝑧 = (𝐽t 𝑘) → ((𝐹𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾)))
4441, 43raleqbidv 3399 . . . . . . . 8 (𝑧 = (𝐽t 𝑘) → (∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾) ↔ ∀𝑔 ∈ ((𝐽t 𝑘) Cn 𝐽)(𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾)))
4544rspcv 3615 . . . . . . 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 4547 . . . . . . . . . . . 12 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
4746adantl 482 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘𝑋)
4847resabs1d 5877 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) = ( I ↾ 𝑘))
49 idcn 21793 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
5049ad3antrrr 726 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
5110ad3antrrr 726 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑋 = 𝐽)
5247, 51sseqtrd 4004 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 𝐽)
536cnrest 21821 . . . . . . . . . . 11 ((( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ 𝑘 𝐽) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
5450, 52, 53syl2anc 584 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
5548, 54eqeltrrd 2911 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
56 coeq2 5722 . . . . . . . . . . 11 (𝑔 = ( I ↾ 𝑘) → (𝐹𝑔) = (𝐹 ∘ ( I ↾ 𝑘)))
5756eleq1d 2894 . . . . . . . . . 10 (𝑔 = ( I ↾ 𝑘) → ((𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽t 𝑘) Cn 𝐾)))
5857rspcv 3615 . . . . . . . . 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 6110 . . . . . . . . 9 (𝐹 ∘ ( I ↾ 𝑘)) = (𝐹𝑘)
6160eleq1i 2900 . . . . . . . 8 ((𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾))
6259, 61syl6ib 252 . . . . . . 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 3186 . . . 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 213 . . 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 579 . 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 280 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933  𝒫 cpw 4535   cuni 4830   I cid 5452  ran crn 5549  cres 5550  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  t crest 16682  TopOnctopon 21446   Cn ccn 21760  Compccmp 21922  𝑘Genckgen 22069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763  df-cmp 21923  df-kgen 22070
This theorem is referenced by: (None)
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