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Theorem kgencn2 23053
Description: A function 𝐹:𝐽⟢𝐾 from a compactly generated space is continuous iff for all compact spaces 𝑧 and continuous 𝑔:π‘§βŸΆπ½, the composite 𝐹 ∘ 𝑔:π‘§βŸΆπΎ is continuous. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
kgencn2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ ((π‘˜Genβ€˜π½) Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘§ ∈ Comp βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))))
Distinct variable groups:   𝑧,𝑔,𝐹   𝑔,𝐽,𝑧   𝑔,𝐾,𝑧   𝑔,𝑋,𝑧   𝑔,π‘Œ,𝑧
Proof of Theorem kgencn2
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 kgencn 23052 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ ((π‘˜Genβ€˜π½) Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)))))
2 rncmp 22892 . . . . . . . 8 ((𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽)) β†’ (𝐽 β†Ύt ran 𝑔) ∈ Comp)
32adantl 483 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ (𝐽 β†Ύt ran 𝑔) ∈ Comp)
4 simprr 772 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ 𝑔 ∈ (𝑧 Cn 𝐽))
5 eqid 2733 . . . . . . . . . . . 12 βˆͺ 𝑧 = βˆͺ 𝑧
6 eqid 2733 . . . . . . . . . . . 12 βˆͺ 𝐽 = βˆͺ 𝐽
75, 6cnf 22742 . . . . . . . . . . 11 (𝑔 ∈ (𝑧 Cn 𝐽) β†’ 𝑔:βˆͺ π‘§βŸΆβˆͺ 𝐽)
8 frn 6722 . . . . . . . . . . 11 (𝑔:βˆͺ π‘§βŸΆβˆͺ 𝐽 β†’ ran 𝑔 βŠ† βˆͺ 𝐽)
94, 7, 83syl 18 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ ran 𝑔 βŠ† βˆͺ 𝐽)
10 toponuni 22408 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1110ad3antrrr 729 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ 𝑋 = βˆͺ 𝐽)
129, 11sseqtrrd 4023 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ ran 𝑔 βŠ† 𝑋)
13 vex 3479 . . . . . . . . . . 11 𝑔 ∈ V
1413rnex 7900 . . . . . . . . . 10 ran 𝑔 ∈ V
1514elpw 4606 . . . . . . . . 9 (ran 𝑔 ∈ 𝒫 𝑋 ↔ ran 𝑔 βŠ† 𝑋)
1612, 15sylibr 233 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ ran 𝑔 ∈ 𝒫 𝑋)
17 oveq2 7414 . . . . . . . . . . 11 (π‘˜ = ran 𝑔 β†’ (𝐽 β†Ύt π‘˜) = (𝐽 β†Ύt ran 𝑔))
1817eleq1d 2819 . . . . . . . . . 10 (π‘˜ = ran 𝑔 β†’ ((𝐽 β†Ύt π‘˜) ∈ Comp ↔ (𝐽 β†Ύt ran 𝑔) ∈ Comp))
19 reseq2 5975 . . . . . . . . . . 11 (π‘˜ = ran 𝑔 β†’ (𝐹 β†Ύ π‘˜) = (𝐹 β†Ύ ran 𝑔))
2017oveq1d 7421 . . . . . . . . . . 11 (π‘˜ = ran 𝑔 β†’ ((𝐽 β†Ύt π‘˜) Cn 𝐾) = ((𝐽 β†Ύt ran 𝑔) Cn 𝐾))
2119, 20eleq12d 2828 . . . . . . . . . 10 (π‘˜ = ran 𝑔 β†’ ((𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾) ↔ (𝐹 β†Ύ ran 𝑔) ∈ ((𝐽 β†Ύt ran 𝑔) Cn 𝐾)))
2218, 21imbi12d 345 . . . . . . . . 9 (π‘˜ = ran 𝑔 β†’ (((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)) ↔ ((𝐽 β†Ύt ran 𝑔) ∈ Comp β†’ (𝐹 β†Ύ ran 𝑔) ∈ ((𝐽 β†Ύt ran 𝑔) Cn 𝐾))))
2322rspcv 3609 . . . . . . . 8 (ran 𝑔 ∈ 𝒫 𝑋 β†’ (βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)) β†’ ((𝐽 β†Ύt ran 𝑔) ∈ Comp β†’ (𝐹 β†Ύ ran 𝑔) ∈ ((𝐽 β†Ύt ran 𝑔) Cn 𝐾))))
2416, 23syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ (βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)) β†’ ((𝐽 β†Ύt ran 𝑔) ∈ Comp β†’ (𝐹 β†Ύ ran 𝑔) ∈ ((𝐽 β†Ύt ran 𝑔) Cn 𝐾))))
253, 24mpid 44 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ (βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)) β†’ (𝐹 β†Ύ ran 𝑔) ∈ ((𝐽 β†Ύt ran 𝑔) Cn 𝐾)))
26 simplll 774 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
27 ssidd 4005 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ ran 𝑔 βŠ† ran 𝑔)
28 cnrest2 22782 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ ran 𝑔 βŠ† ran 𝑔 ∧ ran 𝑔 βŠ† 𝑋) β†’ (𝑔 ∈ (𝑧 Cn 𝐽) ↔ 𝑔 ∈ (𝑧 Cn (𝐽 β†Ύt ran 𝑔))))
2926, 27, 12, 28syl3anc 1372 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ (𝑔 ∈ (𝑧 Cn 𝐽) ↔ 𝑔 ∈ (𝑧 Cn (𝐽 β†Ύt ran 𝑔))))
304, 29mpbid 231 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ 𝑔 ∈ (𝑧 Cn (𝐽 β†Ύt ran 𝑔)))
31 cnco 22762 . . . . . . . . 9 ((𝑔 ∈ (𝑧 Cn (𝐽 β†Ύt ran 𝑔)) ∧ (𝐹 β†Ύ ran 𝑔) ∈ ((𝐽 β†Ύt ran 𝑔) Cn 𝐾)) β†’ ((𝐹 β†Ύ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
3231ex 414 . . . . . . . 8 (𝑔 ∈ (𝑧 Cn (𝐽 β†Ύt ran 𝑔)) β†’ ((𝐹 β†Ύ ran 𝑔) ∈ ((𝐽 β†Ύt ran 𝑔) Cn 𝐾) β†’ ((𝐹 β†Ύ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
3330, 32syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ ((𝐹 β†Ύ ran 𝑔) ∈ ((𝐽 β†Ύt ran 𝑔) Cn 𝐾) β†’ ((𝐹 β†Ύ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
34 ssid 4004 . . . . . . . . 9 ran 𝑔 βŠ† ran 𝑔
35 cores 6246 . . . . . . . . 9 (ran 𝑔 βŠ† ran 𝑔 β†’ ((𝐹 β†Ύ ran 𝑔) ∘ 𝑔) = (𝐹 ∘ 𝑔))
3634, 35ax-mp 5 . . . . . . . 8 ((𝐹 β†Ύ ran 𝑔) ∘ 𝑔) = (𝐹 ∘ 𝑔)
3736eleq1i 2825 . . . . . . 7 (((𝐹 β†Ύ ran 𝑔) ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))
3833, 37imbitrdi 250 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ ((𝐹 β†Ύ ran 𝑔) ∈ ((𝐽 β†Ύt ran 𝑔) Cn 𝐾) β†’ (𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
3925, 38syld 47 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑧 ∈ Comp ∧ 𝑔 ∈ (𝑧 Cn 𝐽))) β†’ (βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)) β†’ (𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
4039ralrimdvva 3210 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)) β†’ βˆ€π‘§ ∈ Comp βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
41 oveq1 7413 . . . . . . . . 9 (𝑧 = (𝐽 β†Ύt π‘˜) β†’ (𝑧 Cn 𝐽) = ((𝐽 β†Ύt π‘˜) Cn 𝐽))
42 oveq1 7413 . . . . . . . . . 10 (𝑧 = (𝐽 β†Ύt π‘˜) β†’ (𝑧 Cn 𝐾) = ((𝐽 β†Ύt π‘˜) Cn 𝐾))
4342eleq2d 2820 . . . . . . . . 9 (𝑧 = (𝐽 β†Ύt π‘˜) β†’ ((𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ (𝐹 ∘ 𝑔) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)))
4441, 43raleqbidv 3343 . . . . . . . 8 (𝑧 = (𝐽 β†Ύt π‘˜) β†’ (βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) ↔ βˆ€π‘” ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)))
4544rspcv 3609 . . . . . . 7 ((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (βˆ€π‘§ ∈ Comp βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) β†’ βˆ€π‘” ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)))
46 elpwi 4609 . . . . . . . . . . . 12 (π‘˜ ∈ 𝒫 𝑋 β†’ π‘˜ βŠ† 𝑋)
4746adantl 483 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ π‘˜ βŠ† 𝑋)
4847resabs1d 6011 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ (( I β†Ύ 𝑋) β†Ύ π‘˜) = ( I β†Ύ π‘˜))
49 idcn 22753 . . . . . . . . . . . 12 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ( I β†Ύ 𝑋) ∈ (𝐽 Cn 𝐽))
5049ad3antrrr 729 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ ( I β†Ύ 𝑋) ∈ (𝐽 Cn 𝐽))
5110ad3antrrr 729 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
5247, 51sseqtrd 4022 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ π‘˜ βŠ† βˆͺ 𝐽)
536cnrest 22781 . . . . . . . . . . 11 ((( I β†Ύ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ π‘˜ βŠ† βˆͺ 𝐽) β†’ (( I β†Ύ 𝑋) β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐽))
5450, 52, 53syl2anc 585 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ (( I β†Ύ 𝑋) β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐽))
5548, 54eqeltrrd 2835 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ ( I β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐽))
56 coeq2 5857 . . . . . . . . . . 11 (𝑔 = ( I β†Ύ π‘˜) β†’ (𝐹 ∘ 𝑔) = (𝐹 ∘ ( I β†Ύ π‘˜)))
5756eleq1d 2819 . . . . . . . . . 10 (𝑔 = ( I β†Ύ π‘˜) β†’ ((𝐹 ∘ 𝑔) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾) ↔ (𝐹 ∘ ( I β†Ύ π‘˜)) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)))
5857rspcv 3609 . . . . . . . . 9 (( I β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐽) β†’ (βˆ€π‘” ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾) β†’ (𝐹 ∘ ( I β†Ύ π‘˜)) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)))
5955, 58syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ (βˆ€π‘” ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾) β†’ (𝐹 ∘ ( I β†Ύ π‘˜)) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)))
60 coires1 6261 . . . . . . . . 9 (𝐹 ∘ ( I β†Ύ π‘˜)) = (𝐹 β†Ύ π‘˜)
6160eleq1i 2825 . . . . . . . 8 ((𝐹 ∘ ( I β†Ύ π‘˜)) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾) ↔ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾))
6259, 61imbitrdi 250 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ (βˆ€π‘” ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐽)(𝐹 ∘ 𝑔) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾) β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)))
6345, 62syl9r 78 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ ((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (βˆ€π‘§ ∈ Comp βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾))))
6463com23 86 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘˜ ∈ 𝒫 𝑋) β†’ (βˆ€π‘§ ∈ Comp βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) β†’ ((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾))))
6564ralrimdva 3155 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘§ ∈ Comp βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾) β†’ βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾))))
6640, 65impbid 211 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾)) ↔ βˆ€π‘§ ∈ Comp βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))
6766pm5.32da 580 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘˜ ∈ 𝒫 𝑋((𝐽 β†Ύt π‘˜) ∈ Comp β†’ (𝐹 β†Ύ π‘˜) ∈ ((𝐽 β†Ύt π‘˜) Cn 𝐾))) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘§ ∈ Comp βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))))
681, 67bitrd 279 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ ((π‘˜Genβ€˜π½) Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘§ ∈ Comp βˆ€π‘” ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908   I cid 5573  ran crn 5677   β†Ύ cres 5678   ∘ ccom 5680  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406   β†Ύt crest 17363  TopOnctopon 22404   Cn ccn 22720  Compccmp 22882  π‘˜Genckgen 23029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-1o 8463  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-fin 8940  df-fi 9403  df-rest 17365  df-topgen 17386  df-top 22388  df-topon 22405  df-bases 22441  df-cn 22723  df-cmp 22883  df-kgen 23030
This theorem is referenced by: (None)
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