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 22005 | . . . 4 ⊢ UnifOn Fn V | |
2 | fnfun 5988 | . . . 4 ⊢ (UnifOn Fn V → Fun UnifOn) | |
3 | elunirn 6509 | . . . 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 22029 | . . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = dom ∪ 𝑈) | |
6 | ustbas.1 | . . . . . . . 8 ⊢ 𝑋 = dom ∪ 𝑈 | |
7 | 5, 6 | syl6eqr 2674 | . . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = 𝑋) |
8 | 7 | fveq2d 6195 | . . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (UnifOn‘𝑥) = (UnifOn‘𝑋)) |
9 | 8 | eleq2d 2687 | . . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋))) |
10 | 9 | ibi 256 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋)) |
11 | 10 | rexlimivw 3029 | . . 3 ⊢ (∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋)) |
12 | 4, 11 | sylbi 207 | . 2 ⊢ (𝑈 ∈ ∪ ran UnifOn → 𝑈 ∈ (UnifOn‘𝑋)) |
13 | elrnust 22028 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn) | |
14 | 12, 13 | impbii 199 | 1 ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ∪ cuni 4436 dom cdm 5114 ran crn 5115 Fun wfun 5882 Fn wfn 5883 ‘cfv 5888 UnifOncust 22003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ust 22004 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |