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Theorem kgencn2 21581
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 21580 . 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 21420 . . . . . . . 8 ((𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽)) → (𝐽t ran 𝑔) ∈ Comp)
32adantl 467 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → (𝐽t ran 𝑔) ∈ Comp)
4 simprr 748 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑔 ∈ (𝑧 Cn 𝐽))
5 eqid 2771 . . . . . . . . . . . 12 𝑧 = 𝑧
6 eqid 2771 . . . . . . . . . . . 12 𝐽 = 𝐽
75, 6cnf 21271 . . . . . . . . . . 11 (𝑔 ∈ (𝑧 Cn 𝐽) → 𝑔: 𝑧 𝐽)
8 frn 6193 . . . . . . . . . . 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 20939 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1110ad3antrrr 701 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝑋 = 𝐽)
129, 11sseqtr4d 3791 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → ran 𝑔𝑋)
13 vex 3354 . . . . . . . . . . 11 𝑔 ∈ V
1413rnex 7247 . . . . . . . . . 10 ran 𝑔 ∈ V
1514elpw 4303 . . . . . . . . 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 6801 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → (𝐽t 𝑘) = (𝐽t ran 𝑔))
1817eleq1d 2835 . . . . . . . . . 10 (𝑘 = ran 𝑔 → ((𝐽t 𝑘) ∈ Comp ↔ (𝐽t ran 𝑔) ∈ Comp))
19 reseq2 5529 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → (𝐹𝑘) = (𝐹 ↾ ran 𝑔))
2017oveq1d 6808 . . . . . . . . . . 11 (𝑘 = ran 𝑔 → ((𝐽t 𝑘) Cn 𝐾) = ((𝐽t ran 𝑔) Cn 𝐾))
2119, 20eleq12d 2844 . . . . . . . . . 10 (𝑘 = ran 𝑔 → ((𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾)))
2218, 21imbi12d 333 . . . . . . . . 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 3456 . . . . . . . 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 750 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) → 𝐽 ∈ (TopOn‘𝑋))
27 ssid 3773 . . . . . . . . . . 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 21311 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran 𝑔 ⊆ ran 𝑔 ∧ ran 𝑔𝑋) → (𝑔 ∈ (𝑧 Cn 𝐽) ↔ 𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔))))
3026, 28, 12, 29syl3anc 1476 . . . . . . . . 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 21291 . . . . . . . . 9 ((𝑔 ∈ (𝑧 Cn (𝐽t ran 𝑔)) ∧ (𝐹 ↾ ran 𝑔) ∈ ((𝐽t ran 𝑔) Cn 𝐾)) → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
3332ex 397 . . . . . . . 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 5782 . . . . . . . . 9 (ran 𝑔 ⊆ ran 𝑔 → ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹𝑔))
3627, 35ax-mp 5 . . . . . . . 8 ((𝐹 ↾ ran 𝑔) ∘ 𝑔) = (𝐹𝑔)
3736eleq1i 2841 . . . . . . 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 3123 . . . 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 6800 . . . . . . . . 9 (𝑧 = (𝐽t 𝑘) → (𝑧 Cn 𝐽) = ((𝐽t 𝑘) Cn 𝐽))
42 oveq1 6800 . . . . . . . . . 10 (𝑧 = (𝐽t 𝑘) → (𝑧 Cn 𝐾) = ((𝐽t 𝑘) Cn 𝐾))
4342eleq2d 2836 . . . . . . . . 9 (𝑧 = (𝐽t 𝑘) → ((𝐹𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾)))
4441, 43raleqbidv 3301 . . . . . . . 8 (𝑧 = (𝐽t 𝑘) → (∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾) ↔ ∀𝑔 ∈ ((𝐽t 𝑘) Cn 𝐽)(𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾)))
4544rspcv 3456 . . . . . . 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 4307 . . . . . . . . . . . 12 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
4746adantl 467 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘𝑋)
4847resabs1d 5569 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) = ( I ↾ 𝑘))
49 idcn 21282 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
5049ad3antrrr 701 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
5110ad3antrrr 701 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑋 = 𝐽)
5247, 51sseqtrd 3790 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 𝐽)
536cnrest 21310 . . . . . . . . . . 11 ((( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ 𝑘 𝐽) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
5450, 52, 53syl2anc 565 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (( I ↾ 𝑋) ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
5548, 54eqeltrrd 2851 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ( I ↾ 𝑘) ∈ ((𝐽t 𝑘) Cn 𝐽))
56 coeq2 5419 . . . . . . . . . . 11 (𝑔 = ( I ↾ 𝑘) → (𝐹𝑔) = (𝐹 ∘ ( I ↾ 𝑘)))
5756eleq1d 2835 . . . . . . . . . 10 (𝑔 = ( I ↾ 𝑘) → ((𝐹𝑔) ∈ ((𝐽t 𝑘) Cn 𝐾) ↔ (𝐹 ∘ ( I ↾ 𝑘)) ∈ ((𝐽t 𝑘) Cn 𝐾)))
5857rspcv 3456 . . . . . . . . 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 5797 . . . . . . . . 9 (𝐹 ∘ ( I ↾ 𝑘)) = (𝐹𝑘)
6160eleq1i 2841 . . . . . . . 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 3118 . . . 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 560 . 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 382   = wceq 1631  wcel 2145  wral 3061  wss 3723  𝒫 cpw 4297   cuni 4574   I cid 5156  ran crn 5250  cres 5251  ccom 5253  wf 6027  cfv 6031  (class class class)co 6793  t crest 16289  TopOnctopon 20935   Cn ccn 21249  Compccmp 21410  𝑘Genckgen 21557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-fin 8113  df-fi 8473  df-rest 16291  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-cn 21252  df-cmp 21411  df-kgen 21558
This theorem is referenced by: (None)
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