| Step | Hyp | Ref
| Expression |
| 1 | | dfac3 10161 |
. 2
⊢
(CHOICE ↔ ∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 2 | | vex 3484 |
. . . . . . 7
⊢ 𝑓 ∈ V |
| 3 | 2 | rnex 7932 |
. . . . . 6
⊢ ran 𝑓 ∈ V |
| 4 | | raleq 3323 |
. . . . . . 7
⊢ (𝑠 = ran 𝑓 → (∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))) |
| 5 | 4 | exbidv 1921 |
. . . . . 6
⊢ (𝑠 = ran 𝑓 → (∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ∃𝑔∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))) |
| 6 | 3, 5 | spcv 3605 |
. . . . 5
⊢
(∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → ∃𝑔∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 7 | | df-nel 3047 |
. . . . . . . . . . . . . . 15
⊢ (∅
∉ ran 𝑓 ↔ ¬
∅ ∈ ran 𝑓) |
| 8 | 7 | biimpi 216 |
. . . . . . . . . . . . . 14
⊢ (∅
∉ ran 𝑓 → ¬
∅ ∈ ran 𝑓) |
| 9 | 8 | ad2antlr 727 |
. . . . . . . . . . . . 13
⊢ (((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ 𝑥 ∈ dom 𝑓) → ¬ ∅ ∈ ran 𝑓) |
| 10 | | fvelrn 7096 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun
𝑓 ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓) |
| 11 | 10 | adantlr 715 |
. . . . . . . . . . . . . . 15
⊢ (((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓) |
| 12 | | eleq1 2829 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓‘𝑥) = ∅ → ((𝑓‘𝑥) ∈ ran 𝑓 ↔ ∅ ∈ ran 𝑓)) |
| 13 | 11, 12 | syl5ibcom 245 |
. . . . . . . . . . . . . 14
⊢ (((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ 𝑥 ∈ dom 𝑓) → ((𝑓‘𝑥) = ∅ → ∅ ∈ ran 𝑓)) |
| 14 | 13 | necon3bd 2954 |
. . . . . . . . . . . . 13
⊢ (((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ 𝑥 ∈ dom 𝑓) → (¬ ∅ ∈ ran 𝑓 → (𝑓‘𝑥) ≠ ∅)) |
| 15 | 9, 14 | mpd 15 |
. . . . . . . . . . . 12
⊢ (((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ≠ ∅) |
| 16 | 15 | adantlr 715 |
. . . . . . . . . . 11
⊢ ((((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ≠ ∅) |
| 17 | | neeq1 3003 |
. . . . . . . . . . . . 13
⊢ (𝑡 = (𝑓‘𝑥) → (𝑡 ≠ ∅ ↔ (𝑓‘𝑥) ≠ ∅)) |
| 18 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (𝑓‘𝑥) → (𝑔‘𝑡) = (𝑔‘(𝑓‘𝑥))) |
| 19 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (𝑓‘𝑥) → 𝑡 = (𝑓‘𝑥)) |
| 20 | 18, 19 | eleq12d 2835 |
. . . . . . . . . . . . 13
⊢ (𝑡 = (𝑓‘𝑥) → ((𝑔‘𝑡) ∈ 𝑡 ↔ (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))) |
| 21 | 17, 20 | imbi12d 344 |
. . . . . . . . . . . 12
⊢ (𝑡 = (𝑓‘𝑥) → ((𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ((𝑓‘𝑥) ≠ ∅ → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))) |
| 22 | | simplr 769 |
. . . . . . . . . . . 12
⊢ ((((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 23 | 10 | ad4ant14 752 |
. . . . . . . . . . . 12
⊢ ((((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓) |
| 24 | 21, 22, 23 | rspcdva 3623 |
. . . . . . . . . . 11
⊢ ((((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → ((𝑓‘𝑥) ≠ ∅ → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))) |
| 25 | 16, 24 | mpd 15 |
. . . . . . . . . 10
⊢ ((((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)) |
| 26 | 25 | ralrimiva 3146 |
. . . . . . . . 9
⊢ (((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)) |
| 27 | 2 | dmex 7931 |
. . . . . . . . . 10
⊢ dom 𝑓 ∈ V |
| 28 | | mptelixpg 8975 |
. . . . . . . . . 10
⊢ (dom
𝑓 ∈ V → ((𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ↔ ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))) |
| 29 | 27, 28 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ↔ ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)) |
| 30 | 26, 29 | sylibr 234 |
. . . . . . . 8
⊢ (((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → (𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥)) |
| 31 | 30 | ne0d 4342 |
. . . . . . 7
⊢ (((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) |
| 32 | 31 | ex 412 |
. . . . . 6
⊢ ((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) → (∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)) |
| 33 | 32 | exlimdv 1933 |
. . . . 5
⊢ ((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) → (∃𝑔∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)) |
| 34 | 6, 33 | syl5com 31 |
. . . 4
⊢
(∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)) |
| 35 | 34 | alrimiv 1927 |
. . 3
⊢
(∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)) |
| 36 | | fnresi 6697 |
. . . . . . 7
⊢ ( I
↾ (𝑠 ∖
{∅})) Fn (𝑠 ∖
{∅}) |
| 37 | | fnfun 6668 |
. . . . . . 7
⊢ (( I
↾ (𝑠 ∖
{∅})) Fn (𝑠 ∖
{∅}) → Fun ( I ↾ (𝑠 ∖ {∅}))) |
| 38 | 36, 37 | ax-mp 5 |
. . . . . 6
⊢ Fun ( I
↾ (𝑠 ∖
{∅})) |
| 39 | | neldifsn 4792 |
. . . . . 6
⊢ ¬
∅ ∈ (𝑠 ∖
{∅}) |
| 40 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑠 ∈ V |
| 41 | 40 | difexi 5330 |
. . . . . . . 8
⊢ (𝑠 ∖ {∅}) ∈
V |
| 42 | | resiexg 7934 |
. . . . . . . 8
⊢ ((𝑠 ∖ {∅}) ∈ V
→ ( I ↾ (𝑠
∖ {∅})) ∈ V) |
| 43 | 41, 42 | ax-mp 5 |
. . . . . . 7
⊢ ( I
↾ (𝑠 ∖
{∅})) ∈ V |
| 44 | | funeq 6586 |
. . . . . . . . 9
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (Fun
𝑓 ↔ Fun ( I ↾
(𝑠 ∖
{∅})))) |
| 45 | | rneq 5947 |
. . . . . . . . . . . . 13
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ran
𝑓 = ran ( I ↾ (𝑠 ∖
{∅}))) |
| 46 | | rnresi 6093 |
. . . . . . . . . . . . 13
⊢ ran ( I
↾ (𝑠 ∖
{∅})) = (𝑠 ∖
{∅}) |
| 47 | 45, 46 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ran
𝑓 = (𝑠 ∖ {∅})) |
| 48 | 47 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
(∅ ∈ ran 𝑓
↔ ∅ ∈ (𝑠
∖ {∅}))) |
| 49 | 48 | notbid 318 |
. . . . . . . . . 10
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
(¬ ∅ ∈ ran 𝑓
↔ ¬ ∅ ∈ (𝑠 ∖ {∅}))) |
| 50 | 7, 49 | bitrid 283 |
. . . . . . . . 9
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
(∅ ∉ ran 𝑓
↔ ¬ ∅ ∈ (𝑠 ∖ {∅}))) |
| 51 | 44, 50 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
((Fun 𝑓 ∧ ∅
∉ ran 𝑓) ↔ (Fun
( I ↾ (𝑠 ∖
{∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅})))) |
| 52 | | dmeq 5914 |
. . . . . . . . . . . 12
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → dom
𝑓 = dom ( I ↾ (𝑠 ∖
{∅}))) |
| 53 | | dmresi 6070 |
. . . . . . . . . . . 12
⊢ dom ( I
↾ (𝑠 ∖
{∅})) = (𝑠 ∖
{∅}) |
| 54 | 52, 53 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → dom
𝑓 = (𝑠 ∖ {∅})) |
| 55 | 54 | ixpeq1d 8949 |
. . . . . . . . . 10
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
X𝑥
∈ dom 𝑓(𝑓‘𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})(𝑓‘𝑥)) |
| 56 | | fveq1 6905 |
. . . . . . . . . . . 12
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
(𝑓‘𝑥) = (( I ↾ (𝑠 ∖ {∅}))‘𝑥)) |
| 57 | | fvresi 7193 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝑠 ∖ {∅}) → (( I ↾
(𝑠 ∖
{∅}))‘𝑥) =
𝑥) |
| 58 | 56, 57 | sylan9eq 2797 |
. . . . . . . . . . 11
⊢ ((𝑓 = ( I ↾ (𝑠 ∖ {∅})) ∧ 𝑥 ∈ (𝑠 ∖ {∅})) → (𝑓‘𝑥) = 𝑥) |
| 59 | 58 | ixpeq2dva 8952 |
. . . . . . . . . 10
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
X𝑥
∈ (𝑠 ∖
{∅})(𝑓‘𝑥) = X𝑥 ∈
(𝑠 ∖ {∅})𝑥) |
| 60 | 55, 59 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
X𝑥
∈ dom 𝑓(𝑓‘𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})𝑥) |
| 61 | 60 | neeq1d 3000 |
. . . . . . . 8
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
(X𝑥
∈ dom 𝑓(𝑓‘𝑥) ≠ ∅ ↔ X𝑥 ∈
(𝑠 ∖ {∅})𝑥 ≠ ∅)) |
| 62 | 51, 61 | imbi12d 344 |
. . . . . . 7
⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) →
(((Fun 𝑓 ∧ ∅
∉ ran 𝑓) → X𝑥 ∈
dom 𝑓(𝑓‘𝑥) ≠ ∅) ↔ ((Fun ( I ↾
(𝑠 ∖ {∅}))
∧ ¬ ∅ ∈ (𝑠 ∖ {∅})) → X𝑥 ∈
(𝑠 ∖ {∅})𝑥 ≠
∅))) |
| 63 | 43, 62 | spcv 3605 |
. . . . . 6
⊢
(∀𝑓((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) → X𝑥 ∈
dom 𝑓(𝑓‘𝑥) ≠ ∅) → ((Fun ( I ↾
(𝑠 ∖ {∅}))
∧ ¬ ∅ ∈ (𝑠 ∖ {∅})) → X𝑥 ∈
(𝑠 ∖ {∅})𝑥 ≠ ∅)) |
| 64 | 38, 39, 63 | mp2ani 698 |
. . . . 5
⊢
(∀𝑓((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) → X𝑥 ∈
dom 𝑓(𝑓‘𝑥) ≠ ∅) → X𝑥 ∈
(𝑠 ∖ {∅})𝑥 ≠ ∅) |
| 65 | | n0 4353 |
. . . . . 6
⊢ (X𝑥 ∈
(𝑠 ∖ {∅})𝑥 ≠ ∅ ↔
∃𝑔 𝑔 ∈ X𝑥 ∈ (𝑠 ∖ {∅})𝑥) |
| 66 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑔 ∈ V |
| 67 | 66 | elixp 8944 |
. . . . . . . 8
⊢ (𝑔 ∈ X𝑥 ∈
(𝑠 ∖ {∅})𝑥 ↔ (𝑔 Fn (𝑠 ∖ {∅}) ∧ ∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥)) |
| 68 | | eldifsn 4786 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝑠 ∖ {∅}) ↔ (𝑥 ∈ 𝑠 ∧ 𝑥 ≠ ∅)) |
| 69 | 68 | imbi1i 349 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ (𝑠 ∖ {∅}) → (𝑔‘𝑥) ∈ 𝑥) ↔ ((𝑥 ∈ 𝑠 ∧ 𝑥 ≠ ∅) → (𝑔‘𝑥) ∈ 𝑥)) |
| 70 | | impexp 450 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ 𝑠 ∧ 𝑥 ≠ ∅) → (𝑔‘𝑥) ∈ 𝑥) ↔ (𝑥 ∈ 𝑠 → (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥))) |
| 71 | 69, 70 | bitri 275 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (𝑠 ∖ {∅}) → (𝑔‘𝑥) ∈ 𝑥) ↔ (𝑥 ∈ 𝑠 → (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥))) |
| 72 | 71 | ralbii2 3089 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
(𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝑠 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) |
| 73 | | neeq1 3003 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑡 → (𝑥 ≠ ∅ ↔ 𝑡 ≠ ∅)) |
| 74 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑡 → (𝑔‘𝑥) = (𝑔‘𝑡)) |
| 75 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑡 → 𝑥 = 𝑡) |
| 76 | 74, 75 | eleq12d 2835 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑡) ∈ 𝑡)) |
| 77 | 73, 76 | imbi12d 344 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑡 → ((𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) ↔ (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))) |
| 78 | 77 | cbvralvw 3237 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
𝑠 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) ↔ ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 79 | 72, 78 | bitri 275 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
(𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥 ↔ ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 80 | 79 | biimpi 216 |
. . . . . . . 8
⊢
(∀𝑥 ∈
(𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥 → ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 81 | 67, 80 | simplbiim 504 |
. . . . . . 7
⊢ (𝑔 ∈ X𝑥 ∈
(𝑠 ∖ {∅})𝑥 → ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 82 | 81 | eximi 1835 |
. . . . . 6
⊢
(∃𝑔 𝑔 ∈ X𝑥 ∈
(𝑠 ∖ {∅})𝑥 → ∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 83 | 65, 82 | sylbi 217 |
. . . . 5
⊢ (X𝑥 ∈
(𝑠 ∖ {∅})𝑥 ≠ ∅ →
∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 84 | 64, 83 | syl 17 |
. . . 4
⊢
(∀𝑓((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) → X𝑥 ∈
dom 𝑓(𝑓‘𝑥) ≠ ∅) → ∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 85 | 84 | alrimiv 1927 |
. . 3
⊢
(∀𝑓((Fun
𝑓 ∧ ∅ ∉ ran
𝑓) → X𝑥 ∈
dom 𝑓(𝑓‘𝑥) ≠ ∅) → ∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) |
| 86 | 35, 85 | impbii 209 |
. 2
⊢
(∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)) |
| 87 | 1, 86 | bitri 275 |
1
⊢
(CHOICE ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)) |