Step | Hyp | Ref
| Expression |
1 | | elex 3441 |
. 2
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) |
2 | | simpll 767 |
. . . . . . . 8
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ 𝑋 ∈ dom
card) |
3 | | elmapi 8553 |
. . . . . . . . . . . 12
⊢ (𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴)
→ 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅})) |
4 | 3 | adantl 485 |
. . . . . . . . . . 11
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅})) |
5 | 4 | frnd 6574 |
. . . . . . . . . 10
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ ran 𝑓 ⊆
(𝒫 𝑋 ∖
{∅})) |
6 | 5 | difss2d 4065 |
. . . . . . . . 9
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ ran 𝑓 ⊆
𝒫 𝑋) |
7 | | sspwuni 5024 |
. . . . . . . . 9
⊢ (ran
𝑓 ⊆ 𝒫 𝑋 ↔ ∪ ran 𝑓 ⊆ 𝑋) |
8 | 6, 7 | sylib 221 |
. . . . . . . 8
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ ∪ ran 𝑓 ⊆ 𝑋) |
9 | | ssnum 9682 |
. . . . . . . 8
⊢ ((𝑋 ∈ dom card ∧ ∪ ran 𝑓 ⊆ 𝑋) → ∪ ran
𝑓 ∈ dom
card) |
10 | 2, 8, 9 | syl2anc 587 |
. . . . . . 7
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ ∪ ran 𝑓 ∈ dom card) |
11 | | ssdifin0 4413 |
. . . . . . . . 9
⊢ (ran
𝑓 ⊆ (𝒫 𝑋 ∖ {∅}) → (ran
𝑓 ∩ {∅}) =
∅) |
12 | 5, 11 | syl 17 |
. . . . . . . 8
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ (ran 𝑓 ∩
{∅}) = ∅) |
13 | | disjsn 4643 |
. . . . . . . 8
⊢ ((ran
𝑓 ∩ {∅}) =
∅ ↔ ¬ ∅ ∈ ran 𝑓) |
14 | 12, 13 | sylib 221 |
. . . . . . 7
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ ¬ ∅ ∈ ran 𝑓) |
15 | | ac5num 9679 |
. . . . . . 7
⊢ ((∪ ran 𝑓 ∈ dom card ∧ ¬ ∅ ∈
ran 𝑓) → ∃ℎ(ℎ:ran 𝑓⟶∪ ran
𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦)) |
16 | 10, 14, 15 | syl2anc 587 |
. . . . . 6
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ ∃ℎ(ℎ:ran 𝑓⟶∪ ran
𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦)) |
17 | | simpllr 776 |
. . . . . . 7
⊢ ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
∧ (ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦)) → 𝐴 ∈ V) |
18 | 4 | ffnd 6567 |
. . . . . . . . . 10
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ 𝑓 Fn 𝐴) |
19 | | fveq2 6738 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑓‘𝑥) → (ℎ‘𝑦) = (ℎ‘(𝑓‘𝑥))) |
20 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑓‘𝑥) → 𝑦 = (𝑓‘𝑥)) |
21 | 19, 20 | eleq12d 2834 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑓‘𝑥) → ((ℎ‘𝑦) ∈ 𝑦 ↔ (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))) |
22 | 21 | ralrn 6928 |
. . . . . . . . . 10
⊢ (𝑓 Fn 𝐴 → (∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))) |
23 | 18, 22 | syl 17 |
. . . . . . . . 9
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ (∀𝑦 ∈
ran 𝑓(ℎ‘𝑦) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))) |
24 | 23 | biimpa 480 |
. . . . . . . 8
⊢ ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
∧ ∀𝑦 ∈ ran
𝑓(ℎ‘𝑦) ∈ 𝑦) → ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)) |
25 | 24 | adantrl 716 |
. . . . . . 7
⊢ ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
∧ (ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦)) → ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)) |
26 | | acnlem 9691 |
. . . . . . 7
⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) |
27 | 17, 25, 26 | syl2anc 587 |
. . . . . 6
⊢ ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
∧ (ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) |
28 | 16, 27 | exlimddv 1943 |
. . . . 5
⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴))
→ ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) |
29 | 28 | ralrimiva 3108 |
. . . 4
⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) |
30 | | isacn 9687 |
. . . 4
⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅})
↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))) |
31 | 29, 30 | mpbird 260 |
. . 3
⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → 𝑋 ∈ AC 𝐴) |
32 | 31 | expcom 417 |
. 2
⊢ (𝐴 ∈ V → (𝑋 ∈ dom card → 𝑋 ∈ AC 𝐴)) |
33 | 1, 32 | syl 17 |
1
⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ dom card → 𝑋 ∈ AC 𝐴)) |