Step |
Hyp |
Ref |
Expression |
1 |
|
mnurndlem1.3 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑈 ) |
2 |
|
mnurndlem1.4 |
⊢ 𝐴 ∈ V |
3 |
|
mnurndlem1.6 |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
4 |
1
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
5 |
|
vex |
⊢ 𝑖 ∈ V |
6 |
5
|
prid1 |
⊢ 𝑖 ∈ { 𝑖 , { ( 𝐹 ‘ 𝑖 ) , 𝐴 } } |
7 |
|
simpr |
⊢ ( ( 𝑖 ∈ 𝐴 ∧ 𝑣 = { 𝑖 , { ( 𝐹 ‘ 𝑖 ) , 𝐴 } } ) → 𝑣 = { 𝑖 , { ( 𝐹 ‘ 𝑖 ) , 𝐴 } } ) |
8 |
6 7
|
eleqtrrid |
⊢ ( ( 𝑖 ∈ 𝐴 ∧ 𝑣 = { 𝑖 , { ( 𝐹 ‘ 𝑖 ) , 𝐴 } } ) → 𝑖 ∈ 𝑣 ) |
9 |
|
eqid |
⊢ ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) = ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) |
10 |
|
id |
⊢ ( 𝑖 ∈ 𝐴 → 𝑖 ∈ 𝐴 ) |
11 |
|
prex |
⊢ { 𝑖 , { ( 𝐹 ‘ 𝑖 ) , 𝐴 } } ∈ V |
12 |
11
|
a1i |
⊢ ( 𝑖 ∈ 𝐴 → { 𝑖 , { ( 𝐹 ‘ 𝑖 ) , 𝐴 } } ∈ V ) |
13 |
|
id |
⊢ ( 𝑎 = 𝑖 → 𝑎 = 𝑖 ) |
14 |
|
fveq2 |
⊢ ( 𝑎 = 𝑖 → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑖 ) ) |
15 |
14
|
preq1d |
⊢ ( 𝑎 = 𝑖 → { ( 𝐹 ‘ 𝑎 ) , 𝐴 } = { ( 𝐹 ‘ 𝑖 ) , 𝐴 } ) |
16 |
13 15
|
preq12d |
⊢ ( 𝑎 = 𝑖 → { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } = { 𝑖 , { ( 𝐹 ‘ 𝑖 ) , 𝐴 } } ) |
17 |
16
|
adantl |
⊢ ( ( 𝑖 ∈ 𝐴 ∧ 𝑎 = 𝑖 ) → { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } = { 𝑖 , { ( 𝐹 ‘ 𝑖 ) , 𝐴 } } ) |
18 |
9 10 12 17
|
rr-elrnmpt3d |
⊢ ( 𝑖 ∈ 𝐴 → { 𝑖 , { ( 𝐹 ‘ 𝑖 ) , 𝐴 } } ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ) |
19 |
8 18
|
rspcime |
⊢ ( 𝑖 ∈ 𝐴 → ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 ) |
20 |
19
|
rgen |
⊢ ∀ 𝑖 ∈ 𝐴 ∃ 𝑣 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) 𝑖 ∈ 𝑣 |
21 |
|
ralim |
⊢ ( ∀ 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 → ∀ 𝑖 ∈ 𝐴 ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
22 |
3 20 21
|
mpisyl |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝐴 ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) |
23 |
|
prex |
⊢ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∈ V |
24 |
23
|
rgenw |
⊢ ∀ 𝑎 ∈ 𝐴 { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∈ V |
25 |
|
eleq2 |
⊢ ( 𝑢 = { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } → ( 𝑖 ∈ 𝑢 ↔ 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ) |
26 |
|
unieq |
⊢ ( 𝑢 = { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } → ∪ 𝑢 = ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) |
27 |
26
|
sseq1d |
⊢ ( 𝑢 = { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } → ( ∪ 𝑢 ⊆ 𝑤 ↔ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) |
28 |
25 27
|
anbi12d |
⊢ ( 𝑢 = { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } → ( ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ↔ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ) |
29 |
9 28
|
rexrnmptw |
⊢ ( ∀ 𝑎 ∈ 𝐴 { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∈ V → ( ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ↔ ∃ 𝑎 ∈ 𝐴 ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ) |
30 |
24 29
|
ax-mp |
⊢ ( ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ↔ ∃ 𝑎 ∈ 𝐴 ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) |
31 |
|
simplrl |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖 ∈ 𝐴 ) → 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) |
32 |
|
simpr |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖 ∈ 𝐴 ) → 𝑖 ∈ 𝐴 ) |
33 |
2
|
prid2 |
⊢ 𝐴 ∈ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } |
34 |
|
elnotel |
⊢ ( 𝐴 ∈ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } → ¬ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ∈ 𝐴 ) |
35 |
33 34
|
ax-mp |
⊢ ¬ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ∈ 𝐴 |
36 |
35
|
a1i |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖 ∈ 𝐴 ) → ¬ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ∈ 𝐴 ) |
37 |
32 36
|
elnelneq2d |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖 ∈ 𝐴 ) → ¬ 𝑖 = { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ) |
38 |
|
elpri |
⊢ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } → ( 𝑖 = 𝑎 ∨ 𝑖 = { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ) ) |
39 |
38
|
orcomd |
⊢ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } → ( 𝑖 = { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ∨ 𝑖 = 𝑎 ) ) |
40 |
39
|
ord |
⊢ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } → ( ¬ 𝑖 = { ( 𝐹 ‘ 𝑎 ) , 𝐴 } → 𝑖 = 𝑎 ) ) |
41 |
31 37 40
|
sylc |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖 ∈ 𝐴 ) → 𝑖 = 𝑎 ) |
42 |
41
|
fveq2d |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑎 ) ) |
43 |
|
simplrr |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖 ∈ 𝐴 ) → ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) |
44 |
|
vex |
⊢ 𝑎 ∈ V |
45 |
|
prex |
⊢ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ∈ V |
46 |
44 45
|
unipr |
⊢ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } = ( 𝑎 ∪ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ) |
47 |
46
|
sseq1i |
⊢ ( ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ↔ ( 𝑎 ∪ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ) ⊆ 𝑤 ) |
48 |
|
unss |
⊢ ( ( 𝑎 ⊆ 𝑤 ∧ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ⊆ 𝑤 ) ↔ ( 𝑎 ∪ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ) ⊆ 𝑤 ) |
49 |
48
|
bicomi |
⊢ ( ( 𝑎 ∪ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ) ⊆ 𝑤 ↔ ( 𝑎 ⊆ 𝑤 ∧ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ⊆ 𝑤 ) ) |
50 |
49
|
simprbi |
⊢ ( ( 𝑎 ∪ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ) ⊆ 𝑤 → { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ⊆ 𝑤 ) |
51 |
47 50
|
sylbi |
⊢ ( ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 → { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ⊆ 𝑤 ) |
52 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑎 ) ∈ V |
53 |
52 2
|
prss |
⊢ ( ( ( 𝐹 ‘ 𝑎 ) ∈ 𝑤 ∧ 𝐴 ∈ 𝑤 ) ↔ { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ⊆ 𝑤 ) |
54 |
53
|
bicomi |
⊢ ( { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ⊆ 𝑤 ↔ ( ( 𝐹 ‘ 𝑎 ) ∈ 𝑤 ∧ 𝐴 ∈ 𝑤 ) ) |
55 |
54
|
simplbi |
⊢ ( { ( 𝐹 ‘ 𝑎 ) , 𝐴 } ⊆ 𝑤 → ( 𝐹 ‘ 𝑎 ) ∈ 𝑤 ) |
56 |
43 51 55
|
3syl |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑎 ) ∈ 𝑤 ) |
57 |
42 56
|
eqeltrd |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑖 ) ∈ 𝑤 ) |
58 |
57
|
ex |
⊢ ( ( 𝑎 ∈ 𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) → ( 𝑖 ∈ 𝐴 → ( 𝐹 ‘ 𝑖 ) ∈ 𝑤 ) ) |
59 |
58
|
rexlimiva |
⊢ ( ∃ 𝑎 ∈ 𝐴 ( 𝑖 ∈ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ∧ ∪ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ⊆ 𝑤 ) → ( 𝑖 ∈ 𝐴 → ( 𝐹 ‘ 𝑖 ) ∈ 𝑤 ) ) |
60 |
30 59
|
sylbi |
⊢ ( ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) → ( 𝑖 ∈ 𝐴 → ( 𝐹 ‘ 𝑖 ) ∈ 𝑤 ) ) |
61 |
60
|
com12 |
⊢ ( 𝑖 ∈ 𝐴 → ( ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) → ( 𝐹 ‘ 𝑖 ) ∈ 𝑤 ) ) |
62 |
61
|
ralimia |
⊢ ( ∀ 𝑖 ∈ 𝐴 ∃ 𝑢 ∈ ran ( 𝑎 ∈ 𝐴 ↦ { 𝑎 , { ( 𝐹 ‘ 𝑎 ) , 𝐴 } } ) ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) → ∀ 𝑖 ∈ 𝐴 ( 𝐹 ‘ 𝑖 ) ∈ 𝑤 ) |
63 |
22 62
|
syl |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝐴 ( 𝐹 ‘ 𝑖 ) ∈ 𝑤 ) |
64 |
|
fnfvrnss |
⊢ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑖 ∈ 𝐴 ( 𝐹 ‘ 𝑖 ) ∈ 𝑤 ) → ran 𝐹 ⊆ 𝑤 ) |
65 |
4 63 64
|
syl2anc |
⊢ ( 𝜑 → ran 𝐹 ⊆ 𝑤 ) |