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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 22809 | . . . 4 ⊢ UnifOn Fn V | |
2 | fnfun 6452 | . . . 4 ⊢ (UnifOn Fn V → Fun UnifOn) | |
3 | elunirn 7009 | . . . 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 22833 | . . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = dom ∪ 𝑈) | |
6 | ustbas.1 | . . . . . . . 8 ⊢ 𝑋 = dom ∪ 𝑈 | |
7 | 5, 6 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = 𝑋) |
8 | 7 | fveq2d 6673 | . . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (UnifOn‘𝑥) = (UnifOn‘𝑋)) |
9 | 8 | eleq2d 2898 | . . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋))) |
10 | 9 | ibi 269 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋)) |
11 | 10 | rexlimivw 3282 | . . 3 ⊢ (∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋)) |
12 | 4, 11 | sylbi 219 | . 2 ⊢ (𝑈 ∈ ∪ ran UnifOn → 𝑈 ∈ (UnifOn‘𝑋)) |
13 | elrnust 22832 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn) | |
14 | 12, 13 | impbii 211 | 1 ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 Vcvv 3494 ∪ cuni 4837 dom cdm 5554 ran crn 5555 Fun wfun 6348 Fn wfn 6349 ‘cfv 6354 UnifOncust 22807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-iota 6313 df-fun 6356 df-fn 6357 df-fv 6362 df-ust 22808 |
This theorem is referenced by: (None) |
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