Step |
Hyp |
Ref |
Expression |
1 |
|
mnurndlem2.1 |
⊢ 𝑀 = { 𝑘 ∣ ∀ 𝑙 ∈ 𝑘 ( 𝒫 𝑙 ⊆ 𝑘 ∧ ∀ 𝑚 ∃ 𝑛 ∈ 𝑘 ( 𝒫 𝑙 ⊆ 𝑛 ∧ ∀ 𝑝 ∈ 𝑙 ( ∃ 𝑞 ∈ 𝑘 ( 𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚 ) → ∃ 𝑟 ∈ 𝑚 ( 𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛 ) ) ) ) } |
2 |
|
mnurndlem2.2 |
⊢ ( 𝜑 → 𝑈 ∈ 𝑀 ) |
3 |
|
mnurndlem2.3 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑈 ) |
4 |
|
mnurndlem2.4 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑈 ) |
5 |
|
mnurndlem2.5 |
⊢ 𝐴 ∈ V |
6 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝑈 ∈ 𝑀 ) |
7 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝐴 ∈ 𝑈 ) |
8 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ∈ 𝐴 ) |
9 |
1 6 7 8
|
mnutrcld |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ∈ 𝑈 ) |
10 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑎 ) ∈ 𝑈 ) |
11 |
1 6 10 7
|
mnuprd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ∈ 𝑈 ) |
12 |
1 6 9 11
|
mnuprd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∈ 𝑈 ) |
13 |
12
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐴 { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∈ 𝑈 ) |
14 |
|
eqid |
⊢ ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) = ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) |
15 |
14
|
rnmptss |
⊢ ( ∀ 𝑎 ∈ 𝐴 { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∈ 𝑈 → ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ⊆ 𝑈 ) |
16 |
13 15
|
syl |
⊢ ( 𝜑 → ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ⊆ 𝑈 ) |
17 |
1 2 3 16
|
mnuop3d |
⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑈 ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
18 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → 𝑤 ∈ 𝑈 ) |
19 |
|
sseq2 |
⊢ ( 𝑏 = 𝑤 → ( ran 𝐹 ⊆ 𝑏 ↔ ran 𝐹 ⊆ 𝑤 ) ) |
20 |
19
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ∧ 𝑏 = 𝑤 ) → ( ran 𝐹 ⊆ 𝑏 ↔ ran 𝐹 ⊆ 𝑤 ) ) |
21 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → 𝐹 : 𝐴 ⟶ 𝑈 ) |
22 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
23 |
21 5 22
|
mnurndlem1 |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → ran 𝐹 ⊆ 𝑤 ) |
24 |
18 20 23
|
rspcedvd |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → ∃ 𝑏 ∈ 𝑈 ran 𝐹 ⊆ 𝑏 ) |
25 |
17 24
|
rexlimddv |
⊢ ( 𝜑 → ∃ 𝑏 ∈ 𝑈 ran 𝐹 ⊆ 𝑏 ) |
26 |
1 2 25
|
mnuss2d |
⊢ ( 𝜑 → ran 𝐹 ∈ 𝑈 ) |