| Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > tposres0 | Structured version Visualization version GIF version | ||
| Description: The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6233 and dftpos6 48869 for an alternate proof. (Contributed by Zhi Wang, 6-Oct-2025.) |
| Ref | Expression |
|---|---|
| tposres0 | ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relres 5956 | . 2 ⊢ Rel (tpos 𝐹 ↾ {∅}) | |
| 2 | relres 5956 | . 2 ⊢ Rel (𝐹 ↾ {∅}) | |
| 3 | velsn 4593 | . . . . 5 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
| 4 | brtpos0 8166 | . . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)) | |
| 5 | 4 | elv 3441 | . . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦) |
| 6 | breq1 5095 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦)) | |
| 7 | breq1 5095 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥𝐹𝑦 ↔ ∅𝐹𝑦)) | |
| 8 | 6, 7 | bibi12d 345 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥tpos 𝐹𝑦 ↔ 𝑥𝐹𝑦) ↔ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))) |
| 9 | 5, 8 | mpbiri 258 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥tpos 𝐹𝑦 ↔ 𝑥𝐹𝑦)) |
| 10 | 3, 9 | sylbi 217 | . . . 4 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ 𝑥𝐹𝑦)) |
| 11 | 10 | pm5.32i 574 | . . 3 ⊢ ((𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦) ↔ (𝑥 ∈ {∅} ∧ 𝑥𝐹𝑦)) |
| 12 | vex 3440 | . . . 4 ⊢ 𝑦 ∈ V | |
| 13 | 12 | brresi 5939 | . . 3 ⊢ (𝑥(tpos 𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦)) |
| 14 | 12 | brresi 5939 | . . 3 ⊢ (𝑥(𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥𝐹𝑦)) |
| 15 | 11, 13, 14 | 3bitr4i 303 | . 2 ⊢ (𝑥(tpos 𝐹 ↾ {∅})𝑦 ↔ 𝑥(𝐹 ↾ {∅})𝑦) |
| 16 | 1, 2, 15 | eqbrriv 5734 | 1 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅}) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3436 ∅c0 4284 {csn 4577 class class class wbr 5092 ↾ cres 5621 tpos ctpos 8158 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-fv 6490 df-tpos 8159 |
| This theorem is referenced by: tposresg 48872 |
| Copyright terms: Public domain | W3C validator |