Step |
Hyp |
Ref |
Expression |
1 |
|
pw2f1o.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
pw2f1o.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
3 |
|
pw2f1o.3 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑊 ) |
4 |
|
pw2f1o.4 |
⊢ ( 𝜑 → 𝐵 ≠ 𝐶 ) |
5 |
|
prid2g |
⊢ ( 𝐶 ∈ 𝑊 → 𝐶 ∈ { 𝐵 , 𝐶 } ) |
6 |
3 5
|
syl |
⊢ ( 𝜑 → 𝐶 ∈ { 𝐵 , 𝐶 } ) |
7 |
|
prid1g |
⊢ ( 𝐵 ∈ 𝑊 → 𝐵 ∈ { 𝐵 , 𝐶 } ) |
8 |
2 7
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ { 𝐵 , 𝐶 } ) |
9 |
6 8
|
ifcld |
⊢ ( 𝜑 → if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ∈ { 𝐵 , 𝐶 } ) |
10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ∈ { 𝐵 , 𝐶 } ) |
11 |
10
|
fmpttd |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) : 𝐴 ⟶ { 𝐵 , 𝐶 } ) |
12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) : 𝐴 ⟶ { 𝐵 , 𝐶 } ) |
13 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) |
14 |
13
|
feq1d |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ↔ ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) : 𝐴 ⟶ { 𝐵 , 𝐶 } ) ) |
15 |
12 14
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ) |
16 |
|
iftrue |
⊢ ( 𝑥 ∈ 𝑆 → if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) = 𝐶 ) |
17 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≠ 𝐶 ) |
18 |
|
iffalse |
⊢ ( ¬ 𝑥 ∈ 𝑆 → if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) = 𝐵 ) |
19 |
18
|
neeq1d |
⊢ ( ¬ 𝑥 ∈ 𝑆 → ( if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 ) ) |
20 |
17 19
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ¬ 𝑥 ∈ 𝑆 → if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) ≠ 𝐶 ) ) |
21 |
20
|
necon4bd |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → ( if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) = 𝐶 → 𝑥 ∈ 𝑆 ) ) |
22 |
16 21
|
impbid2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ∈ 𝑆 ↔ if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) = 𝐶 ) ) |
23 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) |
24 |
23
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) = ( ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ‘ 𝑥 ) ) |
25 |
|
id |
⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴 ) |
26 |
6 8
|
ifcld |
⊢ ( 𝜑 → if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) ∈ { 𝐵 , 𝐶 } ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) ∈ { 𝐵 , 𝐶 } ) |
28 |
|
eleq1w |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆 ) ) |
29 |
28
|
ifbid |
⊢ ( 𝑦 = 𝑥 → if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) = if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
30 |
|
eqid |
⊢ ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
31 |
29 30
|
fvmptg |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) ∈ { 𝐵 , 𝐶 } ) → ( ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ‘ 𝑥 ) = if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
32 |
25 27 31
|
syl2anr |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ‘ 𝑥 ) = if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
33 |
24 32
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) = if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
34 |
33
|
eqeq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑥 ) = 𝐶 ↔ if ( 𝑥 ∈ 𝑆 , 𝐶 , 𝐵 ) = 𝐶 ) ) |
35 |
22 34
|
bitr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ∈ 𝑆 ↔ ( 𝐺 ‘ 𝑥 ) = 𝐶 ) ) |
36 |
35
|
pm5.32da |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑥 ) = 𝐶 ) ) ) |
37 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → 𝑆 ⊆ 𝐴 ) |
38 |
37
|
sseld |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → ( 𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴 ) ) |
39 |
38
|
pm4.71rd |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → ( 𝑥 ∈ 𝑆 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆 ) ) ) |
40 |
|
ffn |
⊢ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } → 𝐺 Fn 𝐴 ) |
41 |
15 40
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → 𝐺 Fn 𝐴 ) |
42 |
|
fniniseg |
⊢ ( 𝐺 Fn 𝐴 → ( 𝑥 ∈ ( ◡ 𝐺 “ { 𝐶 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑥 ) = 𝐶 ) ) ) |
43 |
41 42
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → ( 𝑥 ∈ ( ◡ 𝐺 “ { 𝐶 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑥 ) = 𝐶 ) ) ) |
44 |
36 39 43
|
3bitr4d |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → ( 𝑥 ∈ 𝑆 ↔ 𝑥 ∈ ( ◡ 𝐺 “ { 𝐶 } ) ) ) |
45 |
44
|
eqrdv |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) |
46 |
15 45
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) → ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) |
47 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) |
48 |
|
cnvimass |
⊢ ( ◡ 𝐺 “ { 𝐶 } ) ⊆ dom 𝐺 |
49 |
|
fdm |
⊢ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } → dom 𝐺 = 𝐴 ) |
50 |
49
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → dom 𝐺 = 𝐴 ) |
51 |
48 50
|
sseqtrid |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → ( ◡ 𝐺 “ { 𝐶 } ) ⊆ 𝐴 ) |
52 |
47 51
|
eqsstrd |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → 𝑆 ⊆ 𝐴 ) |
53 |
40
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → 𝐺 Fn 𝐴 ) |
54 |
|
dffn5 |
⊢ ( 𝐺 Fn 𝐴 ↔ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑦 ) ) ) |
55 |
53 54
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → 𝐺 = ( 𝑦 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑦 ) ) ) |
56 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) |
57 |
56
|
eleq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑦 ∈ 𝑆 ↔ 𝑦 ∈ ( ◡ 𝐺 “ { 𝐶 } ) ) ) |
58 |
|
fniniseg |
⊢ ( 𝐺 Fn 𝐴 → ( 𝑦 ∈ ( ◡ 𝐺 “ { 𝐶 } ) ↔ ( 𝑦 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑦 ) = 𝐶 ) ) ) |
59 |
53 58
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → ( 𝑦 ∈ ( ◡ 𝐺 “ { 𝐶 } ) ↔ ( 𝑦 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑦 ) = 𝐶 ) ) ) |
60 |
59
|
baibd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑦 ∈ ( ◡ 𝐺 “ { 𝐶 } ) ↔ ( 𝐺 ‘ 𝑦 ) = 𝐶 ) ) |
61 |
57 60
|
bitrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑦 ∈ 𝑆 ↔ ( 𝐺 ‘ 𝑦 ) = 𝐶 ) ) |
62 |
61
|
biimpa |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑦 ) = 𝐶 ) |
63 |
|
iftrue |
⊢ ( 𝑦 ∈ 𝑆 → if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) = 𝐶 ) |
64 |
63
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝑆 ) → if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) = 𝐶 ) |
65 |
62 64
|
eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
66 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ) |
67 |
66
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑦 ) ∈ { 𝐵 , 𝐶 } ) |
68 |
|
fvex |
⊢ ( 𝐺 ‘ 𝑦 ) ∈ V |
69 |
68
|
elpr |
⊢ ( ( 𝐺 ‘ 𝑦 ) ∈ { 𝐵 , 𝐶 } ↔ ( ( 𝐺 ‘ 𝑦 ) = 𝐵 ∨ ( 𝐺 ‘ 𝑦 ) = 𝐶 ) ) |
70 |
67 69
|
sylib |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑦 ) = 𝐵 ∨ ( 𝐺 ‘ 𝑦 ) = 𝐶 ) ) |
71 |
70
|
ord |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( ¬ ( 𝐺 ‘ 𝑦 ) = 𝐵 → ( 𝐺 ‘ 𝑦 ) = 𝐶 ) ) |
72 |
71 61
|
sylibrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( ¬ ( 𝐺 ‘ 𝑦 ) = 𝐵 → 𝑦 ∈ 𝑆 ) ) |
73 |
72
|
con1d |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( ¬ 𝑦 ∈ 𝑆 → ( 𝐺 ‘ 𝑦 ) = 𝐵 ) ) |
74 |
73
|
imp |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑦 ) = 𝐵 ) |
75 |
|
iffalse |
⊢ ( ¬ 𝑦 ∈ 𝑆 → if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) = 𝐵 ) |
76 |
75
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ∈ 𝑆 ) → if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) = 𝐵 ) |
77 |
74 76
|
eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑦 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
78 |
65 77
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
79 |
78
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → ( 𝑦 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑦 ) ) = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) |
80 |
55 79
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) |
81 |
52 80
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) → ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) |
82 |
46 81
|
impbida |
⊢ ( 𝜑 → ( ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ↔ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ) |
83 |
|
elpw2g |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴 ) ) |
84 |
1 83
|
syl |
⊢ ( 𝜑 → ( 𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴 ) ) |
85 |
|
eleq1w |
⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆 ) ) |
86 |
85
|
ifbid |
⊢ ( 𝑧 = 𝑦 → if ( 𝑧 ∈ 𝑆 , 𝐶 , 𝐵 ) = if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
87 |
86
|
cbvmptv |
⊢ ( 𝑧 ∈ 𝐴 ↦ if ( 𝑧 ∈ 𝑆 , 𝐶 , 𝐵 ) ) = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) |
88 |
87
|
a1i |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐴 ↦ if ( 𝑧 ∈ 𝑆 , 𝐶 , 𝐵 ) ) = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) |
89 |
88
|
eqeq2d |
⊢ ( 𝜑 → ( 𝐺 = ( 𝑧 ∈ 𝐴 ↦ if ( 𝑧 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ↔ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) |
90 |
84 89
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = ( 𝑧 ∈ 𝐴 ↦ if ( 𝑧 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ↔ ( 𝑆 ⊆ 𝐴 ∧ 𝐺 = ( 𝑦 ∈ 𝐴 ↦ if ( 𝑦 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ) ) |
91 |
|
prex |
⊢ { 𝐵 , 𝐶 } ∈ V |
92 |
|
elmapg |
⊢ ( ( { 𝐵 , 𝐶 } ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐺 ∈ ( { 𝐵 , 𝐶 } ↑m 𝐴 ) ↔ 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ) ) |
93 |
91 1 92
|
sylancr |
⊢ ( 𝜑 → ( 𝐺 ∈ ( { 𝐵 , 𝐶 } ↑m 𝐴 ) ↔ 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ) ) |
94 |
93
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝐺 ∈ ( { 𝐵 , 𝐶 } ↑m 𝐴 ) ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ↔ ( 𝐺 : 𝐴 ⟶ { 𝐵 , 𝐶 } ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ) |
95 |
82 90 94
|
3bitr4d |
⊢ ( 𝜑 → ( ( 𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = ( 𝑧 ∈ 𝐴 ↦ if ( 𝑧 ∈ 𝑆 , 𝐶 , 𝐵 ) ) ) ↔ ( 𝐺 ∈ ( { 𝐵 , 𝐶 } ↑m 𝐴 ) ∧ 𝑆 = ( ◡ 𝐺 “ { 𝐶 } ) ) ) ) |