Step | Hyp | Ref
| Expression |
1 | | cnveq 4452 |
. . . . . . . 8
⊢ (x = v →
◡x
= ◡v) |
2 | 1 | eqeq2d 2048 |
. . . . . . 7
⊢ (x = v →
(z = ◡x ↔
z = ◡v)) |
3 | 2 | cbvrexv 2528 |
. . . . . 6
⊢ (∃x ∈ A z = ◡x ↔
∃v ∈ A z = ◡v) |
4 | | cnveq 4452 |
. . . . . . . . . . 11
⊢ (f = v →
◡f
= ◡v) |
5 | 4 | funeqd 4866 |
. . . . . . . . . 10
⊢ (f = v →
(Fun ◡f ↔ Fun ◡v)) |
6 | | sseq1 2960 |
. . . . . . . . . . . 12
⊢ (f = v →
(f ⊆ g ↔ v
⊆ g)) |
7 | | sseq2 2961 |
. . . . . . . . . . . 12
⊢ (f = v →
(g ⊆ f ↔ g
⊆ v)) |
8 | 6, 7 | orbi12d 706 |
. . . . . . . . . . 11
⊢ (f = v →
((f ⊆ g ∨ g ⊆ f)
↔ (v ⊆ g ∨ g ⊆ v))) |
9 | 8 | ralbidv 2320 |
. . . . . . . . . 10
⊢ (f = v →
(∀g
∈ A
(f ⊆ g ∨ g ⊆ f)
↔ ∀g ∈ A (v ⊆
g ∨
g ⊆ v))) |
10 | 5, 9 | anbi12d 442 |
. . . . . . . . 9
⊢ (f = v →
((Fun ◡f ∧ ∀g ∈ A (f ⊆ g
∨ g
⊆ f)) ↔ (Fun ◡v ∧ ∀g ∈ A (v ⊆
g ∨
g ⊆ v)))) |
11 | 10 | rspcv 2646 |
. . . . . . . 8
⊢ (v ∈ A → (∀f ∈ A (Fun ◡f ∧ ∀g ∈ A (f ⊆
g ∨
g ⊆ f)) → (Fun ◡v ∧ ∀g ∈ A (v ⊆
g ∨
g ⊆ v)))) |
12 | | funeq 4864 |
. . . . . . . . . 10
⊢ (z = ◡v →
(Fun z ↔ Fun ◡v)) |
13 | 12 | biimprcd 149 |
. . . . . . . . 9
⊢ (Fun
◡v
→ (z = ◡v →
Fun z)) |
14 | | sseq2 2961 |
. . . . . . . . . . . . . . 15
⊢ (g = x →
(v ⊆ g ↔ v
⊆ x)) |
15 | | sseq1 2960 |
. . . . . . . . . . . . . . 15
⊢ (g = x →
(g ⊆ v ↔ x
⊆ v)) |
16 | 14, 15 | orbi12d 706 |
. . . . . . . . . . . . . 14
⊢ (g = x →
((v ⊆ g ∨ g ⊆ v)
↔ (v ⊆ x ∨ x ⊆ v))) |
17 | 16 | rspcv 2646 |
. . . . . . . . . . . . 13
⊢ (x ∈ A → (∀g ∈ A (v ⊆ g
∨ g
⊆ v) → (v ⊆ x
∨ x
⊆ v))) |
18 | | cnvss 4451 |
. . . . . . . . . . . . . . . 16
⊢ (v ⊆ x
→ ◡v ⊆ ◡x) |
19 | | cnvss 4451 |
. . . . . . . . . . . . . . . 16
⊢ (x ⊆ v
→ ◡x ⊆ ◡v) |
20 | 18, 19 | orim12i 675 |
. . . . . . . . . . . . . . 15
⊢
((v ⊆ x ∨ x ⊆ v)
→ (◡v ⊆ ◡x ∨ ◡x ⊆ ◡v)) |
21 | | sseq12 2962 |
. . . . . . . . . . . . . . . . 17
⊢
((z = ◡v ∧ w = ◡x)
→ (z ⊆ w ↔ ◡v
⊆ ◡x)) |
22 | 21 | ancoms 255 |
. . . . . . . . . . . . . . . 16
⊢
((w = ◡x ∧ z = ◡v)
→ (z ⊆ w ↔ ◡v
⊆ ◡x)) |
23 | | sseq12 2962 |
. . . . . . . . . . . . . . . 16
⊢
((w = ◡x ∧ z = ◡v)
→ (w ⊆ z ↔ ◡x
⊆ ◡v)) |
24 | 22, 23 | orbi12d 706 |
. . . . . . . . . . . . . . 15
⊢
((w = ◡x ∧ z = ◡v)
→ ((z ⊆ w ∨ w ⊆ z)
↔ (◡v ⊆ ◡x ∨ ◡x ⊆ ◡v))) |
25 | 20, 24 | syl5ibrcom 146 |
. . . . . . . . . . . . . 14
⊢
((v ⊆ x ∨ x ⊆ v)
→ ((w = ◡x ∧ z = ◡v)
→ (z ⊆ w ∨ w ⊆ z))) |
26 | 25 | expd 245 |
. . . . . . . . . . . . 13
⊢
((v ⊆ x ∨ x ⊆ v)
→ (w = ◡x →
(z = ◡v →
(z ⊆ w ∨ w ⊆ z)))) |
27 | 17, 26 | syl6com 31 |
. . . . . . . . . . . 12
⊢ (∀g ∈ A (v ⊆ g
∨ g
⊆ v) → (x ∈ A → (w =
◡x
→ (z = ◡v →
(z ⊆ w ∨ w ⊆ z))))) |
28 | 27 | rexlimdv 2426 |
. . . . . . . . . . 11
⊢ (∀g ∈ A (v ⊆ g
∨ g
⊆ v) → (∃x ∈ A w = ◡x →
(z = ◡v →
(z ⊆ w ∨ w ⊆ z)))) |
29 | 28 | com23 72 |
. . . . . . . . . 10
⊢ (∀g ∈ A (v ⊆ g
∨ g
⊆ v) → (z = ◡v →
(∃x
∈ A
w = ◡x →
(z ⊆ w ∨ w ⊆ z)))) |
30 | 29 | alrimdv 1753 |
. . . . . . . . 9
⊢ (∀g ∈ A (v ⊆ g
∨ g
⊆ v) → (z = ◡v →
∀w(∃x ∈ A w = ◡x →
(z ⊆ w ∨ w ⊆ z)))) |
31 | 13, 30 | anim12ii 325 |
. . . . . . . 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 31 |
. . . . . . 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 2426 |
. . . . . 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 | syl5bi 141 |
. . . . 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 1751 |
. . . 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 2305 |
. . . . 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 2554 |
. . . . . . . 8
⊢ z ∈
V |
38 | | eqeq1 2043 |
. . . . . . . . 9
⊢ (y = z →
(y = ◡x ↔
z = ◡x)) |
39 | 38 | rexbidv 2321 |
. . . . . . . 8
⊢ (y = z →
(∃x
∈ A
y = ◡x ↔
∃x ∈ A z = ◡x)) |
40 | 37, 39 | elab 2681 |
. . . . . . 7
⊢ (z ∈ {y ∣ ∃x ∈ A y = ◡x}
↔ ∃x ∈ A z = ◡x) |
41 | | eqeq1 2043 |
. . . . . . . . . 10
⊢ (y = w →
(y = ◡x ↔
w = ◡x)) |
42 | 41 | rexbidv 2321 |
. . . . . . . . 9
⊢ (y = w →
(∃x
∈ A
y = ◡x ↔
∃x ∈ A w = ◡x)) |
43 | 42 | ralab 2695 |
. . . . . . . 8
⊢ (∀w ∈ {y ∣
∃x ∈ A y = ◡x}
(z ⊆ w ∨ w ⊆ z)
↔ ∀w(∃x ∈ A w = ◡x →
(z ⊆ w ∨ w ⊆ z))) |
44 | 43 | anbi2i 430 |
. . . . . . 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 228 |
. . . . . 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 1356 |
. . . . 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 174 |
. . . 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 127 |
. . 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 4910 |
. . 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 14 |
. 2
⊢ (∀f ∈ A (Fun ◡f ∧ ∀g ∈ A (f ⊆
g ∨
g ⊆ f)) → Fun ∪
{y ∣ ∃x ∈ A y = ◡x}) |
51 | | cnvuni 4464 |
. . . 4
⊢ ◡∪ A = ∪ x ∈ A ◡x |
52 | | vex 2554 |
. . . . . 6
⊢ x ∈
V |
53 | 52 | cnvex 4799 |
. . . . 5
⊢ ◡x ∈ V |
54 | 53 | dfiun2 3682 |
. . . 4
⊢ ∪ x ∈ A ◡x =
∪ {y ∣
∃x ∈ A y = ◡x} |
55 | 51, 54 | eqtri 2057 |
. . 3
⊢ ◡∪ A = ∪ {y ∣ ∃x ∈ A y = ◡x} |
56 | 55 | funeqi 4865 |
. 2
⊢ (Fun
◡∪ A ↔ Fun ∪ {y ∣ ∃x ∈ A y = ◡x}) |
57 | 50, 56 | sylibr 137 |
1
⊢ (∀f ∈ A (Fun ◡f ∧ ∀g ∈ A (f ⊆
g ∨
g ⊆ f)) → Fun ◡∪ A) |