Proof of Theorem dfac8clem
| Step | Hyp | Ref
| Expression |
| 1 | | eldifsn 4786 |
. . . . . . 7
⊢ (𝑠 ∈ (𝐴 ∖ {∅}) ↔ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) |
| 2 | | elssuni 4937 |
. . . . . . . . 9
⊢ (𝑠 ∈ 𝐴 → 𝑠 ⊆ ∪ 𝐴) |
| 3 | 2 | ad2antrl 728 |
. . . . . . . 8
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) → 𝑠 ⊆ ∪ 𝐴) |
| 4 | | simplr 769 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) → 𝑟 We ∪ 𝐴) |
| 5 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑟 ∈ V |
| 6 | | exse2 7939 |
. . . . . . . . . . 11
⊢ (𝑟 ∈ V → 𝑟 Se ∪
𝐴) |
| 7 | 5, 6 | mp1i 13 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) → 𝑟 Se ∪ 𝐴) |
| 8 | | simprr 773 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) → 𝑠 ≠ ∅) |
| 9 | | wereu2 5682 |
. . . . . . . . . 10
⊢ (((𝑟 We ∪
𝐴 ∧ 𝑟 Se ∪ 𝐴) ∧ (𝑠 ⊆ ∪ 𝐴 ∧ 𝑠 ≠ ∅)) → ∃!𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎) |
| 10 | 4, 7, 3, 8, 9 | syl22anc 839 |
. . . . . . . . 9
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) → ∃!𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎) |
| 11 | | riotacl 7405 |
. . . . . . . . 9
⊢
(∃!𝑎 ∈
𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎 → (℩𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎) ∈ 𝑠) |
| 12 | 10, 11 | syl 17 |
. . . . . . . 8
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) → (℩𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎) ∈ 𝑠) |
| 13 | 3, 12 | sseldd 3984 |
. . . . . . 7
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) → (℩𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎) ∈ ∪ 𝐴) |
| 14 | 1, 13 | sylan2b 594 |
. . . . . 6
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ 𝑠 ∈ (𝐴 ∖ {∅})) →
(℩𝑎 ∈
𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎) ∈ ∪ 𝐴) |
| 15 | | dfac8clem.1 |
. . . . . 6
⊢ 𝐹 = (𝑠 ∈ (𝐴 ∖ {∅}) ↦
(℩𝑎 ∈
𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎)) |
| 16 | 14, 15 | fmptd 7134 |
. . . . 5
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) → 𝐹:(𝐴 ∖ {∅})⟶∪ 𝐴) |
| 17 | | difexg 5329 |
. . . . . 6
⊢ (𝐴 ∈ 𝐵 → (𝐴 ∖ {∅}) ∈
V) |
| 18 | 17 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) → (𝐴 ∖ {∅}) ∈
V) |
| 19 | | uniexg 7760 |
. . . . . 6
⊢ (𝐴 ∈ 𝐵 → ∪ 𝐴 ∈ V) |
| 20 | 19 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) → ∪ 𝐴
∈ V) |
| 21 | | fex2 7958 |
. . . . 5
⊢ ((𝐹:(𝐴 ∖ {∅})⟶∪ 𝐴
∧ (𝐴 ∖ {∅})
∈ V ∧ ∪ 𝐴 ∈ V) → 𝐹 ∈ V) |
| 22 | 16, 18, 20, 21 | syl3anc 1373 |
. . . 4
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) → 𝐹 ∈ V) |
| 23 | | riotaex 7392 |
. . . . . . . . . . 11
⊢
(℩𝑎
∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎) ∈ V |
| 24 | 15 | fvmpt2 7027 |
. . . . . . . . . . 11
⊢ ((𝑠 ∈ (𝐴 ∖ {∅}) ∧
(℩𝑎 ∈
𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎) ∈ V) → (𝐹‘𝑠) = (℩𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎)) |
| 25 | 23, 24 | mpan2 691 |
. . . . . . . . . 10
⊢ (𝑠 ∈ (𝐴 ∖ {∅}) → (𝐹‘𝑠) = (℩𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎)) |
| 26 | 1, 25 | sylbir 235 |
. . . . . . . . 9
⊢ ((𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅) → (𝐹‘𝑠) = (℩𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎)) |
| 27 | 26 | adantl 481 |
. . . . . . . 8
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) → (𝐹‘𝑠) = (℩𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎)) |
| 28 | 27, 12 | eqeltrd 2841 |
. . . . . . 7
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≠ ∅)) → (𝐹‘𝑠) ∈ 𝑠) |
| 29 | 28 | expr 456 |
. . . . . 6
⊢ (((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) ∧ 𝑠 ∈ 𝐴) → (𝑠 ≠ ∅ → (𝐹‘𝑠) ∈ 𝑠)) |
| 30 | 29 | ralrimiva 3146 |
. . . . 5
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) → ∀𝑠 ∈ 𝐴 (𝑠 ≠ ∅ → (𝐹‘𝑠) ∈ 𝑠)) |
| 31 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑠 𝑧 ≠ ∅ |
| 32 | | nfmpt1 5250 |
. . . . . . . . . 10
⊢
Ⅎ𝑠(𝑠 ∈ (𝐴 ∖ {∅}) ↦
(℩𝑎 ∈
𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎)) |
| 33 | 15, 32 | nfcxfr 2903 |
. . . . . . . . 9
⊢
Ⅎ𝑠𝐹 |
| 34 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑠𝑧 |
| 35 | 33, 34 | nffv 6916 |
. . . . . . . 8
⊢
Ⅎ𝑠(𝐹‘𝑧) |
| 36 | 35 | nfel1 2922 |
. . . . . . 7
⊢
Ⅎ𝑠(𝐹‘𝑧) ∈ 𝑧 |
| 37 | 31, 36 | nfim 1896 |
. . . . . 6
⊢
Ⅎ𝑠(𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) |
| 38 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑧(𝑠 ≠ ∅ → (𝐹‘𝑠) ∈ 𝑠) |
| 39 | | neeq1 3003 |
. . . . . . 7
⊢ (𝑧 = 𝑠 → (𝑧 ≠ ∅ ↔ 𝑠 ≠ ∅)) |
| 40 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑧 = 𝑠 → (𝐹‘𝑧) = (𝐹‘𝑠)) |
| 41 | | id 22 |
. . . . . . . 8
⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠) |
| 42 | 40, 41 | eleq12d 2835 |
. . . . . . 7
⊢ (𝑧 = 𝑠 → ((𝐹‘𝑧) ∈ 𝑧 ↔ (𝐹‘𝑠) ∈ 𝑠)) |
| 43 | 39, 42 | imbi12d 344 |
. . . . . 6
⊢ (𝑧 = 𝑠 → ((𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) ↔ (𝑠 ≠ ∅ → (𝐹‘𝑠) ∈ 𝑠))) |
| 44 | 37, 38, 43 | cbvralw 3306 |
. . . . 5
⊢
(∀𝑧 ∈
𝐴 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) ↔ ∀𝑠 ∈ 𝐴 (𝑠 ≠ ∅ → (𝐹‘𝑠) ∈ 𝑠)) |
| 45 | 30, 44 | sylibr 234 |
. . . 4
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) → ∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)) |
| 46 | | fveq1 6905 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧)) |
| 47 | 46 | eleq1d 2826 |
. . . . . 6
⊢ (𝑓 = 𝐹 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝐹‘𝑧) ∈ 𝑧)) |
| 48 | 47 | imbi2d 340 |
. . . . 5
⊢ (𝑓 = 𝐹 → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))) |
| 49 | 48 | ralbidv 3178 |
. . . 4
⊢ (𝑓 = 𝐹 → (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))) |
| 50 | 22, 45, 49 | spcedv 3598 |
. . 3
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑟 We ∪ 𝐴) → ∃𝑓∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) |
| 51 | 50 | ex 412 |
. 2
⊢ (𝐴 ∈ 𝐵 → (𝑟 We ∪ 𝐴 → ∃𝑓∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))) |
| 52 | 51 | exlimdv 1933 |
1
⊢ (𝐴 ∈ 𝐵 → (∃𝑟 𝑟 We ∪ 𝐴 → ∃𝑓∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))) |