Proof of Theorem suppvalbr
Step | Hyp | Ref
| Expression |
1 | | df-rab 3421 |
. . . 4
⊢ {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})} |
2 | | vex 3467 |
. . . . . . 7
⊢ 𝑥 ∈ V |
3 | 2 | eldm 5898 |
. . . . . 6
⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦) |
4 | | imasng 6084 |
. . . . . . . . 9
⊢ (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦 ∣ 𝑥𝑅𝑦}) |
5 | 4 | elv 3469 |
. . . . . . . 8
⊢ (𝑅 “ {𝑥}) = {𝑦 ∣ 𝑥𝑅𝑦} |
6 | 5 | neeq1i 2995 |
. . . . . . 7
⊢ ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ {𝑦 ∣ 𝑥𝑅𝑦} ≠ {𝑍}) |
7 | | df-sn 4625 |
. . . . . . . 8
⊢ {𝑍} = {𝑦 ∣ 𝑦 = 𝑍} |
8 | 7 | neeq2i 2996 |
. . . . . . 7
⊢ ({𝑦 ∣ 𝑥𝑅𝑦} ≠ {𝑍} ↔ {𝑦 ∣ 𝑥𝑅𝑦} ≠ {𝑦 ∣ 𝑦 = 𝑍}) |
9 | | nabbib 3035 |
. . . . . . 7
⊢ ({𝑦 ∣ 𝑥𝑅𝑦} ≠ {𝑦 ∣ 𝑦 = 𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)) |
10 | 6, 8, 9 | 3bitri 296 |
. . . . . 6
⊢ ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)) |
11 | 3, 10 | anbi12i 626 |
. . . . 5
⊢ ((𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍}) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))) |
12 | 11 | abbii 2796 |
. . . 4
⊢ {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} |
13 | 1, 12 | eqtr2i 2755 |
. . 3
⊢ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} |
14 | 13 | a1i 11 |
. 2
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}}) |
15 | | df-ne 2931 |
. . . . . . 7
⊢ (𝑦 ≠ 𝑍 ↔ ¬ 𝑦 = 𝑍) |
16 | 15 | bibi2i 336 |
. . . . . 6
⊢ ((𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍) ↔ (𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)) |
17 | 16 | exbii 1843 |
. . . . 5
⊢
(∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍) ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)) |
18 | 17 | anbi2i 621 |
. . . 4
⊢
((∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍)) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))) |
19 | 18 | abbii 2796 |
. . 3
⊢ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} |
20 | 19 | a1i 11 |
. 2
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}) |
21 | | suppval 8166 |
. 2
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}}) |
22 | 14, 20, 21 | 3eqtr4rd 2777 |
1
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))}) |