Step |
Hyp |
Ref |
Expression |
1 |
|
cfsetsnfsetfv.f |
⊢ 𝐹 = { 𝑓 ∣ ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑓 ‘ 𝑧 ) = 𝑏 ) } |
2 |
|
cfsetsnfsetfv.g |
⊢ 𝐺 = { 𝑥 ∣ 𝑥 : { 𝑌 } ⟶ 𝐵 } |
3 |
|
cfsetsnfsetfv.h |
⊢ 𝐻 = ( 𝑔 ∈ 𝐺 ↦ ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ) |
4 |
|
simpl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) → 𝐴 ∈ 𝑉 ) |
5 |
4
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → 𝐴 ∈ 𝑉 ) |
6 |
5
|
mptexd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∈ V ) |
7 |
|
vex |
⊢ 𝑔 ∈ V |
8 |
|
feq1 |
⊢ ( 𝑥 = 𝑔 → ( 𝑥 : { 𝑌 } ⟶ 𝐵 ↔ 𝑔 : { 𝑌 } ⟶ 𝐵 ) ) |
9 |
7 8 2
|
elab2 |
⊢ ( 𝑔 ∈ 𝐺 ↔ 𝑔 : { 𝑌 } ⟶ 𝐵 ) |
10 |
9
|
biimpi |
⊢ ( 𝑔 ∈ 𝐺 → 𝑔 : { 𝑌 } ⟶ 𝐵 ) |
11 |
10
|
adantl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → 𝑔 : { 𝑌 } ⟶ 𝐵 ) |
12 |
|
snidg |
⊢ ( 𝑌 ∈ 𝐴 → 𝑌 ∈ { 𝑌 } ) |
13 |
12
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) → 𝑌 ∈ { 𝑌 } ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → 𝑌 ∈ { 𝑌 } ) |
15 |
11 14
|
ffvelrnd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → ( 𝑔 ‘ 𝑌 ) ∈ 𝐵 ) |
16 |
15
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) ∧ 𝑎 ∈ 𝐴 ) → ( 𝑔 ‘ 𝑌 ) ∈ 𝐵 ) |
17 |
16
|
fmpttd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) : 𝐴 ⟶ 𝐵 ) |
18 |
|
eqeq2 |
⊢ ( 𝑏 = ( 𝑔 ‘ 𝑌 ) → ( ( 𝑔 ‘ 𝑌 ) = 𝑏 ↔ ( 𝑔 ‘ 𝑌 ) = ( 𝑔 ‘ 𝑌 ) ) ) |
19 |
18
|
ralbidv |
⊢ ( 𝑏 = ( 𝑔 ‘ 𝑌 ) → ( ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = 𝑏 ↔ ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = ( 𝑔 ‘ 𝑌 ) ) ) |
20 |
19
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) ∧ 𝑏 = ( 𝑔 ‘ 𝑌 ) ) → ( ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = 𝑏 ↔ ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = ( 𝑔 ‘ 𝑌 ) ) ) |
21 |
|
eqidd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑔 ‘ 𝑌 ) = ( 𝑔 ‘ 𝑌 ) ) |
22 |
21
|
ralrimiva |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = ( 𝑔 ‘ 𝑌 ) ) |
23 |
15 20 22
|
rspcedvd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = 𝑏 ) |
24 |
17 23
|
jca |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → ( ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) : 𝐴 ⟶ 𝐵 ∧ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = 𝑏 ) ) |
25 |
|
feq1 |
⊢ ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) → ( 𝑓 : 𝐴 ⟶ 𝐵 ↔ ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) : 𝐴 ⟶ 𝐵 ) ) |
26 |
|
simpl |
⊢ ( ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∧ 𝑧 ∈ 𝐴 ) → 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ) |
27 |
|
eqidd |
⊢ ( ( ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑎 = 𝑧 ) → ( 𝑔 ‘ 𝑌 ) = ( 𝑔 ‘ 𝑌 ) ) |
28 |
|
simpr |
⊢ ( ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ 𝐴 ) |
29 |
|
fvexd |
⊢ ( ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑔 ‘ 𝑌 ) ∈ V ) |
30 |
|
nfcv |
⊢ Ⅎ 𝑎 𝑓 |
31 |
|
nfmpt1 |
⊢ Ⅎ 𝑎 ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) |
32 |
30 31
|
nfeq |
⊢ Ⅎ 𝑎 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) |
33 |
|
nfv |
⊢ Ⅎ 𝑎 𝑧 ∈ 𝐴 |
34 |
32 33
|
nfan |
⊢ Ⅎ 𝑎 ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∧ 𝑧 ∈ 𝐴 ) |
35 |
|
nfcv |
⊢ Ⅎ 𝑎 𝑧 |
36 |
|
nfcv |
⊢ Ⅎ 𝑎 ( 𝑔 ‘ 𝑌 ) |
37 |
26 27 28 29 34 35 36
|
fvmptdf |
⊢ ( ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑓 ‘ 𝑧 ) = ( 𝑔 ‘ 𝑌 ) ) |
38 |
37
|
eqeq1d |
⊢ ( ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑓 ‘ 𝑧 ) = 𝑏 ↔ ( 𝑔 ‘ 𝑌 ) = 𝑏 ) ) |
39 |
38
|
ralbidva |
⊢ ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) → ( ∀ 𝑧 ∈ 𝐴 ( 𝑓 ‘ 𝑧 ) = 𝑏 ↔ ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = 𝑏 ) ) |
40 |
39
|
rexbidv |
⊢ ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) → ( ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑓 ‘ 𝑧 ) = 𝑏 ↔ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = 𝑏 ) ) |
41 |
25 40
|
anbi12d |
⊢ ( 𝑓 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) → ( ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑓 ‘ 𝑧 ) = 𝑏 ) ↔ ( ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) : 𝐴 ⟶ 𝐵 ∧ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑔 ‘ 𝑌 ) = 𝑏 ) ) ) |
42 |
6 24 41
|
elabd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∈ { 𝑓 ∣ ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑓 ‘ 𝑧 ) = 𝑏 ) } ) |
43 |
42 1
|
eleqtrrdi |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐺 ) → ( 𝑎 ∈ 𝐴 ↦ ( 𝑔 ‘ 𝑌 ) ) ∈ 𝐹 ) |
44 |
43 3
|
fmptd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) → 𝐻 : 𝐺 ⟶ 𝐹 ) |