| Step | Hyp | Ref
| Expression |
| 1 | | cnveq 5884 |
. . . . . . . 8
⊢ (𝑥 = 𝑣 → ◡𝑥 = ◡𝑣) |
| 2 | 1 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝑥 = 𝑣 → (𝑧 = ◡𝑥 ↔ 𝑧 = ◡𝑣)) |
| 3 | 2 | cbvrexvw 3238 |
. . . . . 6
⊢
(∃𝑥 ∈
𝐴 𝑧 = ◡𝑥 ↔ ∃𝑣 ∈ 𝐴 𝑧 = ◡𝑣) |
| 4 | | cnveq 5884 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑣 → ◡𝑓 = ◡𝑣) |
| 5 | 4 | funeqd 6588 |
. . . . . . . . . 10
⊢ (𝑓 = 𝑣 → (Fun ◡𝑓 ↔ Fun ◡𝑣)) |
| 6 | | sseq1 4009 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑣 → (𝑓 ⊆ 𝑔 ↔ 𝑣 ⊆ 𝑔)) |
| 7 | | sseq2 4010 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑣 → (𝑔 ⊆ 𝑓 ↔ 𝑔 ⊆ 𝑣)) |
| 8 | 6, 7 | orbi12d 919 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑣 → ((𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓) ↔ (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣))) |
| 9 | 8 | ralbidv 3178 |
. . . . . . . . . 10
⊢ (𝑓 = 𝑣 → (∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓) ↔ ∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣))) |
| 10 | 5, 9 | anbi12d 632 |
. . . . . . . . 9
⊢ (𝑓 = 𝑣 → ((Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ↔ (Fun ◡𝑣 ∧ ∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣)))) |
| 11 | 10 | rspcv 3618 |
. . . . . . . 8
⊢ (𝑣 ∈ 𝐴 → (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ◡𝑣 ∧ ∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣)))) |
| 12 | | funeq 6586 |
. . . . . . . . . 10
⊢ (𝑧 = ◡𝑣 → (Fun 𝑧 ↔ Fun ◡𝑣)) |
| 13 | 12 | biimprcd 250 |
. . . . . . . . 9
⊢ (Fun
◡𝑣 → (𝑧 = ◡𝑣 → Fun 𝑧)) |
| 14 | | sseq2 4010 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝑥 → (𝑣 ⊆ 𝑔 ↔ 𝑣 ⊆ 𝑥)) |
| 15 | | sseq1 4009 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝑥 → (𝑔 ⊆ 𝑣 ↔ 𝑥 ⊆ 𝑣)) |
| 16 | 14, 15 | orbi12d 919 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = 𝑥 → ((𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) ↔ (𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣))) |
| 17 | 16 | rspcv 3618 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝐴 → (∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣))) |
| 18 | | cnvss 5883 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 ⊆ 𝑥 → ◡𝑣 ⊆ ◡𝑥) |
| 19 | | cnvss 5883 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ⊆ 𝑣 → ◡𝑥 ⊆ ◡𝑣) |
| 20 | 18, 19 | orim12i 909 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣) → (◡𝑣 ⊆ ◡𝑥 ∨ ◡𝑥 ⊆ ◡𝑣)) |
| 21 | | sseq12 4011 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑧 = ◡𝑣 ∧ 𝑤 = ◡𝑥) → (𝑧 ⊆ 𝑤 ↔ ◡𝑣 ⊆ ◡𝑥)) |
| 22 | 21 | ancoms 458 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑤 = ◡𝑥 ∧ 𝑧 = ◡𝑣) → (𝑧 ⊆ 𝑤 ↔ ◡𝑣 ⊆ ◡𝑥)) |
| 23 | | sseq12 4011 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑤 = ◡𝑥 ∧ 𝑧 = ◡𝑣) → (𝑤 ⊆ 𝑧 ↔ ◡𝑥 ⊆ ◡𝑣)) |
| 24 | 22, 23 | orbi12d 919 |
. . . . . . . . . . . . . . 15
⊢ ((𝑤 = ◡𝑥 ∧ 𝑧 = ◡𝑣) → ((𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧) ↔ (◡𝑣 ⊆ ◡𝑥 ∨ ◡𝑥 ⊆ ◡𝑣))) |
| 25 | 20, 24 | syl5ibrcom 247 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣) → ((𝑤 = ◡𝑥 ∧ 𝑧 = ◡𝑣) → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))) |
| 26 | 25 | expd 415 |
. . . . . . . . . . . . 13
⊢ ((𝑣 ⊆ 𝑥 ∨ 𝑥 ⊆ 𝑣) → (𝑤 = ◡𝑥 → (𝑧 = ◡𝑣 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))) |
| 27 | 17, 26 | syl6com 37 |
. . . . . . . . . . . 12
⊢
(∀𝑔 ∈
𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (𝑥 ∈ 𝐴 → (𝑤 = ◡𝑥 → (𝑧 = ◡𝑣 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))) |
| 28 | 27 | rexlimdv 3153 |
. . . . . . . . . . 11
⊢
(∀𝑔 ∈
𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 = ◡𝑣 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))) |
| 29 | 28 | com23 86 |
. . . . . . . . . 10
⊢
(∀𝑔 ∈
𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (𝑧 = ◡𝑣 → (∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))) |
| 30 | 29 | alrimdv 1929 |
. . . . . . . . 9
⊢
(∀𝑔 ∈
𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣) → (𝑧 = ◡𝑣 → ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))) |
| 31 | 13, 30 | anim12ii 618 |
. . . . . . . 8
⊢ ((Fun
◡𝑣 ∧ ∀𝑔 ∈ 𝐴 (𝑣 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑣)) → (𝑧 = ◡𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))) |
| 32 | 11, 31 | syl6com 37 |
. . . . . . 7
⊢
(∀𝑓 ∈
𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (𝑣 ∈ 𝐴 → (𝑧 = ◡𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))))) |
| 33 | 32 | rexlimdv 3153 |
. . . . . 6
⊢
(∀𝑓 ∈
𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (∃𝑣 ∈ 𝐴 𝑧 = ◡𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))) |
| 34 | 3, 33 | biimtrid 242 |
. . . . 5
⊢
(∀𝑓 ∈
𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))) |
| 35 | 34 | alrimiv 1927 |
. . . 4
⊢
(∀𝑓 ∈
𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))) |
| 36 | | df-ral 3062 |
. . . . 5
⊢
(∀𝑧 ∈
{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)) ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} → (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))) |
| 37 | | vex 3484 |
. . . . . . . 8
⊢ 𝑧 ∈ V |
| 38 | | eqeq1 2741 |
. . . . . . . . 9
⊢ (𝑦 = 𝑧 → (𝑦 = ◡𝑥 ↔ 𝑧 = ◡𝑥)) |
| 39 | 38 | rexbidv 3179 |
. . . . . . . 8
⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥)) |
| 40 | 37, 39 | elab 3679 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} ↔ ∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥) |
| 41 | | eqeq1 2741 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑤 → (𝑦 = ◡𝑥 ↔ 𝑤 = ◡𝑥)) |
| 42 | 41 | rexbidv 3179 |
. . . . . . . . 9
⊢ (𝑦 = 𝑤 → (∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥)) |
| 43 | 42 | ralab 3697 |
. . . . . . . 8
⊢
(∀𝑤 ∈
{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧) ↔ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))) |
| 44 | 43 | anbi2i 623 |
. . . . . . 7
⊢ ((Fun
𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)) ↔ (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))) |
| 45 | 40, 44 | imbi12i 350 |
. . . . . 6
⊢ ((𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} → (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))) ↔ (∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))) |
| 46 | 45 | albii 1819 |
. . . . 5
⊢
(∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} → (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))))) |
| 47 | 36, 46 | bitr2i 276 |
. . . 4
⊢
(∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = ◡𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥 ∈ 𝐴 𝑤 = ◡𝑥 → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))) ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))) |
| 48 | 35, 47 | sylib 218 |
. . 3
⊢
(∀𝑓 ∈
𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))) |
| 49 | | fununi 6641 |
. . 3
⊢
(∀𝑧 ∈
{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)) → Fun ∪
{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥}) |
| 50 | 48, 49 | syl 17 |
. 2
⊢
(∀𝑓 ∈
𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪
{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥}) |
| 51 | | cnvuni 5897 |
. . . 4
⊢ ◡∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ◡𝑥 |
| 52 | | vex 3484 |
. . . . . 6
⊢ 𝑥 ∈ V |
| 53 | 52 | cnvex 7947 |
. . . . 5
⊢ ◡𝑥 ∈ V |
| 54 | 53 | dfiun2 5033 |
. . . 4
⊢ ∪ 𝑥 ∈ 𝐴 ◡𝑥 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} |
| 55 | 51, 54 | eqtri 2765 |
. . 3
⊢ ◡∪ 𝐴 = ∪
{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ◡𝑥} |
| 56 | 55 | funeqi 6587 |
. 2
⊢ (Fun
◡∪ 𝐴 ↔ Fun ∪ {𝑦
∣ ∃𝑥 ∈
𝐴 𝑦 = ◡𝑥}) |
| 57 | 50, 56 | sylibr 234 |
1
⊢
(∀𝑓 ∈
𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ◡∪ 𝐴) |