|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 24211 | . . . 4 ⊢ UnifOn Fn V | |
| 2 | fnfun 6667 | . . . 4 ⊢ (UnifOn Fn V → Fun UnifOn) | |
| 3 | elunirn 7272 | . . . 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 24235 | . . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = dom ∪ 𝑈) | |
| 6 | ustbas.1 | . . . . . . . 8 ⊢ 𝑋 = dom ∪ 𝑈 | |
| 7 | 5, 6 | eqtr4di 2794 | . . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = 𝑋) | 
| 8 | 7 | fveq2d 6909 | . . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (UnifOn‘𝑥) = (UnifOn‘𝑋)) | 
| 9 | 8 | eleq2d 2826 | . . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋))) | 
| 10 | 9 | ibi 267 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋)) | 
| 11 | 10 | rexlimivw 3150 | . . 3 ⊢ (∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋)) | 
| 12 | 4, 11 | sylbi 217 | . 2 ⊢ (𝑈 ∈ ∪ ran UnifOn → 𝑈 ∈ (UnifOn‘𝑋)) | 
| 13 | elfvunirn 6937 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn) | |
| 14 | 12, 13 | impbii 209 | 1 ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 Vcvv 3479 ∪ cuni 4906 dom cdm 5684 ran crn 5685 Fun wfun 6554 Fn wfn 6555 ‘cfv 6560 UnifOncust 24209 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 df-ust 24210 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |