Step |
Hyp |
Ref |
Expression |
1 |
|
frege133.x |
⊢ 𝑋 ∈ 𝑈 |
2 |
|
frege133.y |
⊢ 𝑌 ∈ 𝑉 |
3 |
|
frege133.m |
⊢ 𝑀 ∈ 𝑊 |
4 |
|
frege133.r |
⊢ 𝑅 ∈ 𝑆 |
5 |
|
fvex |
⊢ ( t+ ‘ 𝑅 ) ∈ V |
6 |
5
|
cnvex |
⊢ ◡ ( t+ ‘ 𝑅 ) ∈ V |
7 |
|
imaexg |
⊢ ( ◡ ( t+ ‘ 𝑅 ) ∈ V → ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∈ V ) |
8 |
6 7
|
ax-mp |
⊢ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∈ V |
9 |
|
imaundir |
⊢ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) = ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( I “ { 𝑀 } ) ) |
10 |
|
imaexg |
⊢ ( ( t+ ‘ 𝑅 ) ∈ V → ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∈ V ) |
11 |
5 10
|
ax-mp |
⊢ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∈ V |
12 |
|
imai |
⊢ ( I “ { 𝑀 } ) = { 𝑀 } |
13 |
|
snex |
⊢ { 𝑀 } ∈ V |
14 |
12 13
|
eqeltri |
⊢ ( I “ { 𝑀 } ) ∈ V |
15 |
11 14
|
unex |
⊢ ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( I “ { 𝑀 } ) ) ∈ V |
16 |
9 15
|
eqeltri |
⊢ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ∈ V |
17 |
1 2 4 8 16
|
frege83 |
⊢ ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ) ) |
18 |
3
|
elexi |
⊢ 𝑀 ∈ V |
19 |
1
|
elexi |
⊢ 𝑋 ∈ V |
20 |
18 19
|
elimasn |
⊢ ( 𝑋 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ 〈 𝑀 , 𝑋 〉 ∈ ◡ ( t+ ‘ 𝑅 ) ) |
21 |
|
df-br |
⊢ ( 𝑀 ◡ ( t+ ‘ 𝑅 ) 𝑋 ↔ 〈 𝑀 , 𝑋 〉 ∈ ◡ ( t+ ‘ 𝑅 ) ) |
22 |
18 19
|
brcnv |
⊢ ( 𝑀 ◡ ( t+ ‘ 𝑅 ) 𝑋 ↔ 𝑋 ( t+ ‘ 𝑅 ) 𝑀 ) |
23 |
20 21 22
|
3bitr2i |
⊢ ( 𝑋 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ 𝑋 ( t+ ‘ 𝑅 ) 𝑀 ) |
24 |
|
elun |
⊢ ( 𝑌 ∈ ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ↔ ( 𝑌 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∨ 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) |
25 |
|
df-or |
⊢ ( ( 𝑌 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∨ 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ↔ ( ¬ 𝑌 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) |
26 |
2
|
elexi |
⊢ 𝑌 ∈ V |
27 |
18 26
|
elimasn |
⊢ ( 𝑌 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ 〈 𝑀 , 𝑌 〉 ∈ ◡ ( t+ ‘ 𝑅 ) ) |
28 |
|
df-br |
⊢ ( 𝑀 ◡ ( t+ ‘ 𝑅 ) 𝑌 ↔ 〈 𝑀 , 𝑌 〉 ∈ ◡ ( t+ ‘ 𝑅 ) ) |
29 |
18 26
|
brcnv |
⊢ ( 𝑀 ◡ ( t+ ‘ 𝑅 ) 𝑌 ↔ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 ) |
30 |
27 28 29
|
3bitr2i |
⊢ ( 𝑌 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 ) |
31 |
30
|
notbii |
⊢ ( ¬ 𝑌 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 ) |
32 |
18 26
|
elimasn |
⊢ ( 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ↔ 〈 𝑀 , 𝑌 〉 ∈ ( ( t+ ‘ 𝑅 ) ∪ I ) ) |
33 |
|
df-br |
⊢ ( 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ↔ 〈 𝑀 , 𝑌 〉 ∈ ( ( t+ ‘ 𝑅 ) ∪ I ) ) |
34 |
32 33
|
bitr4i |
⊢ ( 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ↔ 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) |
35 |
31 34
|
imbi12i |
⊢ ( ( ¬ 𝑌 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ↔ ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) |
36 |
24 25 35
|
3bitri |
⊢ ( 𝑌 ∈ ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ↔ ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) |
37 |
36
|
imbi2i |
⊢ ( ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ↔ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) |
38 |
23 37
|
imbi12i |
⊢ ( ( 𝑋 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ) ↔ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) |
39 |
38
|
imbi2i |
⊢ ( ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ) ) ↔ ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) ) |
40 |
3 4
|
frege132 |
⊢ ( ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) ) |
41 |
39 40
|
sylbi |
⊢ ( ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ∈ ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) ) |
42 |
17 41
|
ax-mp |
⊢ ( Fun ◡ ◡ 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) |