Step | Hyp | Ref
| Expression |
1 | | difss 4022 |
. . . . 5
⊢ (𝐹 ∖ 𝐺) ⊆ 𝐹 |
2 | | dmss 5745 |
. . . . 5
⊢ ((𝐹 ∖ 𝐺) ⊆ 𝐹 → dom (𝐹 ∖ 𝐺) ⊆ dom 𝐹) |
3 | 1, 2 | ax-mp 5 |
. . . 4
⊢ dom
(𝐹 ∖ 𝐺) ⊆ dom 𝐹 |
4 | | fndm 6440 |
. . . . 5
⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) |
5 | 4 | adantr 484 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom 𝐹 = 𝐴) |
6 | 3, 5 | sseqtrid 3929 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) ⊆ 𝐴) |
7 | | sseqin2 4106 |
. . 3
⊢ (dom
(𝐹 ∖ 𝐺) ⊆ 𝐴 ↔ (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺)) |
8 | 6, 7 | sylib 221 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = dom (𝐹 ∖ 𝐺)) |
9 | | vex 3402 |
. . . . 5
⊢ 𝑥 ∈ V |
10 | 9 | eldm 5743 |
. . . 4
⊢ (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦) |
11 | | eqcom 2745 |
. . . . . . . . 9
⊢ ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥)) |
12 | | fnbrfvb 6724 |
. . . . . . . . 9
⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥))) |
13 | 11, 12 | syl5bb 286 |
. . . . . . . 8
⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥))) |
14 | 13 | adantll 714 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ 𝑥𝐺(𝐹‘𝑥))) |
15 | 14 | necon3abid 2970 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ¬ 𝑥𝐺(𝐹‘𝑥))) |
16 | | fvex 6689 |
. . . . . . 7
⊢ (𝐹‘𝑥) ∈ V |
17 | | breq2 5034 |
. . . . . . . 8
⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐺𝑦 ↔ 𝑥𝐺(𝐹‘𝑥))) |
18 | 17 | notbid 321 |
. . . . . . 7
⊢ (𝑦 = (𝐹‘𝑥) → (¬ 𝑥𝐺𝑦 ↔ ¬ 𝑥𝐺(𝐹‘𝑥))) |
19 | 16, 18 | ceqsexv 3445 |
. . . . . 6
⊢
(∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ¬ 𝑥𝐺(𝐹‘𝑥)) |
20 | 15, 19 | bitr4di 292 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦))) |
21 | | eqcom 2745 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) |
22 | | fnbrfvb 6724 |
. . . . . . . . . 10
⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) |
23 | 21, 22 | syl5bb 286 |
. . . . . . . . 9
⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)) |
24 | 23 | adantlr 715 |
. . . . . . . 8
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)) |
25 | 24 | anbi1d 633 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦))) |
26 | | brdif 5083 |
. . . . . . 7
⊢ (𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐺𝑦)) |
27 | 25, 26 | bitr4di 292 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ 𝑥(𝐹 ∖ 𝐺)𝑦)) |
28 | 27 | exbidv 1928 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ ¬ 𝑥𝐺𝑦) ↔ ∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦)) |
29 | 20, 28 | bitr2d 283 |
. . . 4
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑦 𝑥(𝐹 ∖ 𝐺)𝑦 ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥))) |
30 | 10, 29 | syl5bb 286 |
. . 3
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ 𝐺) ↔ (𝐹‘𝑥) ≠ (𝐺‘𝑥))) |
31 | 30 | rabbi2dva 4108 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐴 ∩ dom (𝐹 ∖ 𝐺)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)}) |
32 | 8, 31 | eqtr3d 2775 |
1
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)}) |