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Mirrors > Home > MPE Home > Th. List > ustbas | Structured version Visualization version GIF version |
Description: Recover the base of an uniform structure 𝑈. ∪ ran UnifOn is to UnifOn what Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
Ref | Expression |
---|---|
ustbas.1 | ⊢ 𝑋 = dom ∪ 𝑈 |
Ref | Expression |
---|---|
ustbas | ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ustfn 22807 | . . . 4 ⊢ UnifOn Fn V | |
2 | fnfun 6423 | . . . 4 ⊢ (UnifOn Fn V → Fun UnifOn) | |
3 | elunirn 6988 | . . . 4 ⊢ (Fun UnifOn → (𝑈 ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))) | |
4 | 1, 2, 3 | mp2b 10 | . . 3 ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥)) |
5 | ustbas2 22831 | . . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = dom ∪ 𝑈) | |
6 | ustbas.1 | . . . . . . . 8 ⊢ 𝑋 = dom ∪ 𝑈 | |
7 | 5, 6 | eqtr4di 2851 | . . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = 𝑋) |
8 | 7 | fveq2d 6649 | . . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (UnifOn‘𝑥) = (UnifOn‘𝑋)) |
9 | 8 | eleq2d 2875 | . . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋))) |
10 | 9 | ibi 270 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋)) |
11 | 10 | rexlimivw 3241 | . . 3 ⊢ (∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋)) |
12 | 4, 11 | sylbi 220 | . 2 ⊢ (𝑈 ∈ ∪ ran UnifOn → 𝑈 ∈ (UnifOn‘𝑋)) |
13 | elrnust 22830 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn) | |
14 | 12, 13 | impbii 212 | 1 ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∪ cuni 4800 dom cdm 5519 ran crn 5520 Fun wfun 6318 Fn wfn 6319 ‘cfv 6324 UnifOncust 22805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 df-ust 22806 |
This theorem is referenced by: (None) |
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