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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmsrcl | Structured version Visualization version GIF version |
Description: Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
cvmcov.1 | ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) |
Ref | Expression |
---|---|
cvmsrcl | ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmcov.1 | . . 3 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) | |
2 | 1 | dmmptss 6095 | . 2 ⊢ dom 𝑆 ⊆ 𝐽 |
3 | elfvdm 6702 | . 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ dom 𝑆) | |
4 | 2, 3 | sseldi 3965 | 1 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 ∖ cdif 3933 ∩ cin 3935 ∅c0 4291 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 ↦ cmpt 5146 ◡ccnv 5554 dom cdm 5555 ↾ cres 5557 “ cima 5558 ‘cfv 6355 (class class class)co 7156 ↾t crest 16694 Homeochmeo 22361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fv 6363 |
This theorem is referenced by: cvmsi 32512 cvmsf1o 32519 cvmsss2 32521 cvmopnlem 32525 cvmliftlem8 32539 cvmlift2lem9 32558 cvmlift2lem10 32559 cvmlift3lem6 32571 cvmlift3lem8 32573 |
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