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 |
1 2 3
|
cfsetsnfsetf |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) → 𝐻 : 𝐺 ⟶ 𝐹 ) |
5 |
|
vex |
⊢ 𝑚 ∈ V |
6 |
|
feq1 |
⊢ ( 𝑓 = 𝑚 → ( 𝑓 : 𝐴 ⟶ 𝐵 ↔ 𝑚 : 𝐴 ⟶ 𝐵 ) ) |
7 |
|
fveq1 |
⊢ ( 𝑓 = 𝑚 → ( 𝑓 ‘ 𝑧 ) = ( 𝑚 ‘ 𝑧 ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴 ) → ( 𝑓 ‘ 𝑧 ) = ( 𝑚 ‘ 𝑧 ) ) |
9 |
8
|
eqeq1d |
⊢ ( ( 𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑓 ‘ 𝑧 ) = 𝑏 ↔ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ) |
10 |
9
|
ralbidva |
⊢ ( 𝑓 = 𝑚 → ( ∀ 𝑧 ∈ 𝐴 ( 𝑓 ‘ 𝑧 ) = 𝑏 ↔ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ) |
11 |
10
|
rexbidv |
⊢ ( 𝑓 = 𝑚 → ( ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑓 ‘ 𝑧 ) = 𝑏 ↔ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ) |
12 |
6 11
|
anbi12d |
⊢ ( 𝑓 = 𝑚 → ( ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑓 ‘ 𝑧 ) = 𝑏 ) ↔ ( 𝑚 : 𝐴 ⟶ 𝐵 ∧ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ) ) |
13 |
5 12 1
|
elab2 |
⊢ ( 𝑚 ∈ 𝐹 ↔ ( 𝑚 : 𝐴 ⟶ 𝐵 ∧ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ) |
14 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ∧ 𝑦 ∈ { 𝑌 } ) → 𝑏 ∈ 𝐵 ) |
15 |
|
eqid |
⊢ ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) = ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) |
16 |
14 15
|
fmptd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) : { 𝑌 } ⟶ 𝐵 ) |
17 |
|
snex |
⊢ { 𝑌 } ∈ V |
18 |
17
|
mptex |
⊢ ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ∈ V |
19 |
|
feq1 |
⊢ ( 𝑥 = ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) → ( 𝑥 : { 𝑌 } ⟶ 𝐵 ↔ ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) : { 𝑌 } ⟶ 𝐵 ) ) |
20 |
18 19 2
|
elab2 |
⊢ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ∈ 𝐺 ↔ ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) : { 𝑌 } ⟶ 𝐵 ) |
21 |
16 20
|
sylibr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ∈ 𝐺 ) |
22 |
|
fveq1 |
⊢ ( 𝑛 = ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) → ( 𝑛 ‘ 𝑌 ) = ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) |
23 |
22
|
mpteq2dv |
⊢ ( 𝑛 = ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) → ( 𝑎 ∈ 𝐴 ↦ ( 𝑛 ‘ 𝑌 ) ) = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ) |
24 |
23
|
eqeq2d |
⊢ ( 𝑛 = ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) → ( 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑛 ‘ 𝑌 ) ) ↔ 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ) ) |
25 |
24
|
adantl |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ∧ 𝑛 = ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ) → ( 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑛 ‘ 𝑌 ) ) ↔ 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ) ) |
26 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ( 𝑚 ‘ 𝑧 ) = 𝑏 ) |
27 |
|
eqidd |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ) |
28 |
|
eqidd |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ∧ 𝑎 = 𝑧 ) → ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) = ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ) |
29 |
|
eqidd |
⊢ ( ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ∧ 𝑎 = 𝑧 ) ∧ 𝑦 = 𝑌 ) → 𝑏 = 𝑏 ) |
30 |
|
snidg |
⊢ ( 𝑌 ∈ 𝐴 → 𝑌 ∈ { 𝑌 } ) |
31 |
30
|
ad6antlr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ∧ 𝑎 = 𝑧 ) → 𝑌 ∈ { 𝑌 } ) |
32 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → 𝑏 ∈ 𝐵 ) |
33 |
32
|
adantr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ∧ 𝑎 = 𝑧 ) → 𝑏 ∈ 𝐵 ) |
34 |
28 29 31 33
|
fvmptd |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ∧ 𝑎 = 𝑧 ) → ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) = 𝑏 ) |
35 |
|
simpr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ 𝐴 ) |
36 |
35
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → 𝑧 ∈ 𝐴 ) |
37 |
27 34 36 32
|
fvmptd |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ( ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ‘ 𝑧 ) = 𝑏 ) |
38 |
26 37
|
eqtr4d |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) ∧ ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ( 𝑚 ‘ 𝑧 ) = ( ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ‘ 𝑧 ) ) |
39 |
38
|
ex |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑚 ‘ 𝑧 ) = 𝑏 → ( 𝑚 ‘ 𝑧 ) = ( ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ‘ 𝑧 ) ) ) |
40 |
39
|
ralimdva |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) → ( ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 → ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = ( ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ‘ 𝑧 ) ) ) |
41 |
40
|
imp |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = ( ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ‘ 𝑧 ) ) |
42 |
|
ffn |
⊢ ( 𝑚 : 𝐴 ⟶ 𝐵 → 𝑚 Fn 𝐴 ) |
43 |
42
|
adantl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) → 𝑚 Fn 𝐴 ) |
44 |
|
nfv |
⊢ Ⅎ 𝑎 ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) |
45 |
|
fvexd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑎 ∈ 𝐴 ) → ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ∈ V ) |
46 |
|
eqid |
⊢ ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) |
47 |
44 45 46
|
fnmptd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) → ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) Fn 𝐴 ) |
48 |
43 47
|
jca |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) → ( 𝑚 Fn 𝐴 ∧ ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) Fn 𝐴 ) ) |
49 |
48
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) → ( 𝑚 Fn 𝐴 ∧ ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) Fn 𝐴 ) ) |
50 |
49
|
adantr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ( 𝑚 Fn 𝐴 ∧ ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) Fn 𝐴 ) ) |
51 |
|
eqfnfv |
⊢ ( ( 𝑚 Fn 𝐴 ∧ ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) Fn 𝐴 ) → ( 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ↔ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = ( ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ‘ 𝑧 ) ) ) |
52 |
50 51
|
syl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ( 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ↔ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = ( ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ‘ 𝑧 ) ) ) |
53 |
41 52
|
mpbird |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( ( 𝑦 ∈ { 𝑌 } ↦ 𝑏 ) ‘ 𝑌 ) ) ) |
54 |
21 25 53
|
rspcedvd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑛 ‘ 𝑌 ) ) ) |
55 |
|
simp-4l |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → 𝐴 ∈ 𝑉 ) |
56 |
1 2 3
|
cfsetsnfsetfv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑛 ∈ 𝐺 ) → ( 𝐻 ‘ 𝑛 ) = ( 𝑎 ∈ 𝐴 ↦ ( 𝑛 ‘ 𝑌 ) ) ) |
57 |
55 56
|
sylan |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ∧ 𝑛 ∈ 𝐺 ) → ( 𝐻 ‘ 𝑛 ) = ( 𝑎 ∈ 𝐴 ↦ ( 𝑛 ‘ 𝑌 ) ) ) |
58 |
57
|
eqeq2d |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) ∧ 𝑛 ∈ 𝐺 ) → ( 𝑚 = ( 𝐻 ‘ 𝑛 ) ↔ 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑛 ‘ 𝑌 ) ) ) ) |
59 |
58
|
rexbidva |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ( ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝐻 ‘ 𝑛 ) ↔ ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝑎 ∈ 𝐴 ↦ ( 𝑛 ‘ 𝑌 ) ) ) ) |
60 |
54 59
|
mpbird |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝐻 ‘ 𝑛 ) ) |
61 |
60
|
ex |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) ∧ 𝑏 ∈ 𝐵 ) → ( ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 → ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝐻 ‘ 𝑛 ) ) ) |
62 |
61
|
rexlimdva |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) ∧ 𝑚 : 𝐴 ⟶ 𝐵 ) → ( ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 → ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝐻 ‘ 𝑛 ) ) ) |
63 |
62
|
expimpd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝑚 : 𝐴 ⟶ 𝐵 ∧ ∃ 𝑏 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( 𝑚 ‘ 𝑧 ) = 𝑏 ) → ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝐻 ‘ 𝑛 ) ) ) |
64 |
13 63
|
syl5bi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑚 ∈ 𝐹 → ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝐻 ‘ 𝑛 ) ) ) |
65 |
64
|
ralrimiv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) → ∀ 𝑚 ∈ 𝐹 ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝐻 ‘ 𝑛 ) ) |
66 |
|
dffo3 |
⊢ ( 𝐻 : 𝐺 –onto→ 𝐹 ↔ ( 𝐻 : 𝐺 ⟶ 𝐹 ∧ ∀ 𝑚 ∈ 𝐹 ∃ 𝑛 ∈ 𝐺 𝑚 = ( 𝐻 ‘ 𝑛 ) ) ) |
67 |
4 65 66
|
sylanbrc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴 ) → 𝐻 : 𝐺 –onto→ 𝐹 ) |