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Theorem xkoopn 23085
Description: A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
xkoopn.x 𝑋 = βˆͺ 𝑅
xkoopn.r (πœ‘ β†’ 𝑅 ∈ Top)
xkoopn.s (πœ‘ β†’ 𝑆 ∈ Top)
xkoopn.a (πœ‘ β†’ 𝐴 βŠ† 𝑋)
xkoopn.c (πœ‘ β†’ (𝑅 β†Ύt 𝐴) ∈ Comp)
xkoopn.u (πœ‘ β†’ π‘ˆ ∈ 𝑆)
Assertion
Ref Expression
xkoopn (πœ‘ β†’ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} ∈ (𝑆 ↑ko 𝑅))
Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑆,𝑓   π‘ˆ,𝑓
Allowed substitution hints:   πœ‘(𝑓)   𝑋(𝑓)
Proof of Theorem xkoopn
Dummy variables π‘˜ 𝑣 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7439 . . . . . . 7 (𝑅 Cn 𝑆) ∈ V
21pwex 5378 . . . . . 6 𝒫 (𝑅 Cn 𝑆) ∈ V
3 xkoopn.x . . . . . . . 8 𝑋 = βˆͺ 𝑅
4 eqid 2733 . . . . . . . 8 {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp} = {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}
5 eqid 2733 . . . . . . . 8 (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) = (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})
63, 4, 5xkotf 23081 . . . . . . 7 (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}):({π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp} Γ— 𝑆)βŸΆπ’« (𝑅 Cn 𝑆)
7 frn 6722 . . . . . . 7 ((π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}):({π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp} Γ— 𝑆)βŸΆπ’« (𝑅 Cn 𝑆) β†’ ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) βŠ† 𝒫 (𝑅 Cn 𝑆))
86, 7ax-mp 5 . . . . . 6 ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) βŠ† 𝒫 (𝑅 Cn 𝑆)
92, 8ssexi 5322 . . . . 5 ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) ∈ V
10 ssfii 9411 . . . . 5 (ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) ∈ V β†’ ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) βŠ† (fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})))
119, 10ax-mp 5 . . . 4 ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) βŠ† (fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))
12 fvex 6902 . . . . 5 (fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})) ∈ V
13 bastg 22461 . . . . 5 ((fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})) ∈ V β†’ (fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})) βŠ† (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))))
1412, 13ax-mp 5 . . . 4 (fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})) βŠ† (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})))
1511, 14sstri 3991 . . 3 ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) βŠ† (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})))
16 oveq2 7414 . . . . . . 7 (π‘₯ = 𝐴 β†’ (𝑅 β†Ύt π‘₯) = (𝑅 β†Ύt 𝐴))
1716eleq1d 2819 . . . . . 6 (π‘₯ = 𝐴 β†’ ((𝑅 β†Ύt π‘₯) ∈ Comp ↔ (𝑅 β†Ύt 𝐴) ∈ Comp))
18 xkoopn.a . . . . . . 7 (πœ‘ β†’ 𝐴 βŠ† 𝑋)
19 xkoopn.r . . . . . . . 8 (πœ‘ β†’ 𝑅 ∈ Top)
203topopn 22400 . . . . . . . 8 (𝑅 ∈ Top β†’ 𝑋 ∈ 𝑅)
21 elpw2g 5344 . . . . . . . 8 (𝑋 ∈ 𝑅 β†’ (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 βŠ† 𝑋))
2219, 20, 213syl 18 . . . . . . 7 (πœ‘ β†’ (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 βŠ† 𝑋))
2318, 22mpbird 257 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝒫 𝑋)
24 xkoopn.c . . . . . 6 (πœ‘ β†’ (𝑅 β†Ύt 𝐴) ∈ Comp)
2517, 23, 24elrabd 3685 . . . . 5 (πœ‘ β†’ 𝐴 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp})
26 xkoopn.u . . . . 5 (πœ‘ β†’ π‘ˆ ∈ 𝑆)
27 eqidd 2734 . . . . 5 (πœ‘ β†’ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ})
28 imaeq2 6054 . . . . . . . . 9 (π‘˜ = 𝐴 β†’ (𝑓 β€œ π‘˜) = (𝑓 β€œ 𝐴))
2928sseq1d 4013 . . . . . . . 8 (π‘˜ = 𝐴 β†’ ((𝑓 β€œ π‘˜) βŠ† 𝑣 ↔ (𝑓 β€œ 𝐴) βŠ† 𝑣))
3029rabbidv 3441 . . . . . . 7 (π‘˜ = 𝐴 β†’ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† 𝑣})
3130eqeq2d 2744 . . . . . 6 (π‘˜ = 𝐴 β†’ ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣} ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† 𝑣}))
32 sseq2 4008 . . . . . . . 8 (𝑣 = π‘ˆ β†’ ((𝑓 β€œ 𝐴) βŠ† 𝑣 ↔ (𝑓 β€œ 𝐴) βŠ† π‘ˆ))
3332rabbidv 3441 . . . . . . 7 (𝑣 = π‘ˆ β†’ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ})
3433eqeq2d 2744 . . . . . 6 (𝑣 = π‘ˆ β†’ ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† 𝑣} ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ}))
3531, 34rspc2ev 3624 . . . . 5 ((𝐴 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp} ∧ π‘ˆ ∈ 𝑆 ∧ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ}) β†’ βˆƒπ‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}βˆƒπ‘£ ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})
3625, 26, 27, 35syl3anc 1372 . . . 4 (πœ‘ β†’ βˆƒπ‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}βˆƒπ‘£ ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})
371rabex 5332 . . . . 5 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} ∈ V
38 eqeq1 2737 . . . . . 6 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} β†’ (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣} ↔ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))
39382rexbidv 3220 . . . . 5 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} β†’ (βˆƒπ‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}βˆƒπ‘£ ∈ 𝑆 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣} ↔ βˆƒπ‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}βˆƒπ‘£ ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))
405rnmpo 7539 . . . . 5 ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) = {𝑦 ∣ βˆƒπ‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}βˆƒπ‘£ ∈ 𝑆 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}}
4137, 39, 40elab2 3672 . . . 4 ({𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} ∈ ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}) ↔ βˆƒπ‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}βˆƒπ‘£ ∈ 𝑆 {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣})
4236, 41sylibr 233 . . 3 (πœ‘ β†’ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} ∈ ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))
4315, 42sselid 3980 . 2 (πœ‘ β†’ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} ∈ (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))))
44 xkoopn.s . . 3 (πœ‘ β†’ 𝑆 ∈ Top)
453, 4, 5xkoval 23083 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝑅) = (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))))
4619, 44, 45syl2anc 585 . 2 (πœ‘ β†’ (𝑆 ↑ko 𝑅) = (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑅 β†Ύt π‘₯) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ π‘˜) βŠ† 𝑣}))))
4743, 46eleqtrrd 2837 1 (πœ‘ β†’ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 β€œ 𝐴) βŠ† π‘ˆ} ∈ (𝑆 ↑ko 𝑅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908   Γ— cxp 5674  ran crn 5677   β€œ cima 5679  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406   ∈ cmpo 7408  ficfi 9402   β†Ύt crest 17363  topGenctg 17380  Topctop 22387   Cn ccn 22720  Compccmp 22882   ↑ko cxko 23057
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-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-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-en 8937  df-fin 8940  df-fi 9403  df-topgen 17386  df-top 22388  df-xko 23059
This theorem is referenced by:  xkouni  23095  xkohaus  23149  xkoptsub  23150  xkoco1cn  23153  xkoco2cn  23154  xkococnlem  23155
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