Step |
Hyp |
Ref |
Expression |
1 |
|
zorn2lem.3 |
⊢ 𝐹 = recs ( ( 𝑓 ∈ V ↦ ( ℩ 𝑣 ∈ 𝐶 ∀ 𝑢 ∈ 𝐶 ¬ 𝑢 𝑤 𝑣 ) ) ) |
2 |
|
zorn2lem.4 |
⊢ 𝐶 = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ran 𝑓 𝑔 𝑅 𝑧 } |
3 |
|
zorn2lem.5 |
⊢ 𝐷 = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑥 ) 𝑔 𝑅 𝑧 } |
4 |
|
zorn2lem.7 |
⊢ 𝐻 = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } |
5 |
1
|
tfr1 |
⊢ 𝐹 Fn On |
6 |
|
fnfun |
⊢ ( 𝐹 Fn On → Fun 𝐹 ) |
7 |
5 6
|
ax-mp |
⊢ Fun 𝐹 |
8 |
|
fvelima |
⊢ ( ( Fun 𝐹 ∧ 𝑠 ∈ ( 𝐹 “ 𝑥 ) ) → ∃ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑠 ) |
9 |
7 8
|
mpan |
⊢ ( 𝑠 ∈ ( 𝐹 “ 𝑥 ) → ∃ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑠 ) |
10 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) |
11 |
|
nfra1 |
⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ |
12 |
10 11
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) |
13 |
|
nfv |
⊢ Ⅎ 𝑦 𝑠 ∈ 𝐴 |
14 |
|
df-ral |
⊢ ( ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝐻 ≠ ∅ ) ) |
15 |
|
onelon |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ On ) |
16 |
4
|
ssrab3 |
⊢ 𝐻 ⊆ 𝐴 |
17 |
1 2 4
|
zorn2lem1 |
⊢ ( ( 𝑦 ∈ On ∧ ( 𝑤 We 𝐴 ∧ 𝐻 ≠ ∅ ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝐻 ) |
18 |
16 17
|
sselid |
⊢ ( ( 𝑦 ∈ On ∧ ( 𝑤 We 𝐴 ∧ 𝐻 ≠ ∅ ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝐴 ) |
19 |
|
eleq1 |
⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝐴 ↔ 𝑠 ∈ 𝐴 ) ) |
20 |
18 19
|
syl5ib |
⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → ( ( 𝑦 ∈ On ∧ ( 𝑤 We 𝐴 ∧ 𝐻 ≠ ∅ ) ) → 𝑠 ∈ 𝐴 ) ) |
21 |
15 20
|
sylani |
⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → ( ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑤 We 𝐴 ∧ 𝐻 ≠ ∅ ) ) → 𝑠 ∈ 𝐴 ) ) |
22 |
21
|
com12 |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑤 We 𝐴 ∧ 𝐻 ≠ ∅ ) ) → ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → 𝑠 ∈ 𝐴 ) ) |
23 |
22
|
exp43 |
⊢ ( 𝑥 ∈ On → ( 𝑦 ∈ 𝑥 → ( 𝑤 We 𝐴 → ( 𝐻 ≠ ∅ → ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → 𝑠 ∈ 𝐴 ) ) ) ) ) |
24 |
23
|
com3r |
⊢ ( 𝑤 We 𝐴 → ( 𝑥 ∈ On → ( 𝑦 ∈ 𝑥 → ( 𝐻 ≠ ∅ → ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → 𝑠 ∈ 𝐴 ) ) ) ) ) |
25 |
24
|
imp |
⊢ ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) → ( 𝑦 ∈ 𝑥 → ( 𝐻 ≠ ∅ → ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → 𝑠 ∈ 𝐴 ) ) ) ) |
26 |
25
|
a2d |
⊢ ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) → ( ( 𝑦 ∈ 𝑥 → 𝐻 ≠ ∅ ) → ( 𝑦 ∈ 𝑥 → ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → 𝑠 ∈ 𝐴 ) ) ) ) |
27 |
26
|
spsd |
⊢ ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) → ( ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝐻 ≠ ∅ ) → ( 𝑦 ∈ 𝑥 → ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → 𝑠 ∈ 𝐴 ) ) ) ) |
28 |
14 27
|
syl5bi |
⊢ ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) → ( ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ → ( 𝑦 ∈ 𝑥 → ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → 𝑠 ∈ 𝐴 ) ) ) ) |
29 |
28
|
imp |
⊢ ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ( 𝑦 ∈ 𝑥 → ( ( 𝐹 ‘ 𝑦 ) = 𝑠 → 𝑠 ∈ 𝐴 ) ) ) |
30 |
12 13 29
|
rexlimd |
⊢ ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ( ∃ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑠 → 𝑠 ∈ 𝐴 ) ) |
31 |
9 30
|
syl5 |
⊢ ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ( 𝑠 ∈ ( 𝐹 “ 𝑥 ) → 𝑠 ∈ 𝐴 ) ) |
32 |
31
|
ssrdv |
⊢ ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ( 𝐹 “ 𝑥 ) ⊆ 𝐴 ) |