| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ordsson |
⊢ ( Ord 𝐴 → 𝐴 ⊆ On ) |
| 2 |
1
|
3ad2ant2 |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → 𝐴 ⊆ On ) |
| 3 |
2
|
sseld |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ On ) ) |
| 4 |
|
eleq1w |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) ) |
| 5 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) |
| 6 |
|
id |
⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 ) |
| 7 |
5 6
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ‘ 𝑥 ) = 𝑥 ↔ ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) |
| 8 |
4 7
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ↔ ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ) |
| 9 |
8
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ↔ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ) ) |
| 10 |
|
r19.21v |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ↔ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ) |
| 11 |
|
ordelss |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ⊆ 𝐴 ) |
| 12 |
11
|
3ad2antl2 |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ⊆ 𝐴 ) |
| 13 |
12
|
sselda |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝐴 ) |
| 14 |
|
pm5.5 |
⊢ ( 𝑦 ∈ 𝐴 → ( ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ↔ ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) |
| 15 |
13 14
|
syl |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ↔ ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) |
| 16 |
15
|
ralbidva |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) |
| 17 |
|
isof1o |
⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |
| 18 |
17
|
3ad2ant1 |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |
| 19 |
18
|
ad2antrr |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |
| 20 |
|
simpll3 |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → Ord 𝐵 ) |
| 21 |
|
simpr |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) |
| 22 |
|
f1of |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
| 23 |
17 22
|
syl |
⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
| 24 |
23
|
3ad2ant1 |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
| 25 |
24
|
ad2antrr |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
| 26 |
|
simplrl |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → 𝑥 ∈ 𝐴 ) |
| 27 |
25 26
|
ffvelrnd |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) |
| 28 |
21 27
|
jca |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ( 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) ) |
| 29 |
|
ordtr1 |
⊢ ( Ord 𝐵 → ( ( 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) → 𝑧 ∈ 𝐵 ) ) |
| 30 |
20 28 29
|
sylc |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → 𝑧 ∈ 𝐵 ) |
| 31 |
|
f1ocnvfv2 |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) = 𝑧 ) |
| 32 |
19 30 31
|
syl2anc |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) = 𝑧 ) |
| 33 |
32 21
|
eqeltrd |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) ∈ ( 𝐹 ‘ 𝑥 ) ) |
| 34 |
|
simpll1 |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ) |
| 35 |
|
f1ocnv |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ◡ 𝐹 : 𝐵 –1-1-onto→ 𝐴 ) |
| 36 |
|
f1of |
⊢ ( ◡ 𝐹 : 𝐵 –1-1-onto→ 𝐴 → ◡ 𝐹 : 𝐵 ⟶ 𝐴 ) |
| 37 |
19 35 36
|
3syl |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ◡ 𝐹 : 𝐵 ⟶ 𝐴 ) |
| 38 |
37 30
|
ffvelrnd |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ( ◡ 𝐹 ‘ 𝑧 ) ∈ 𝐴 ) |
| 39 |
|
isorel |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ ( ( ◡ 𝐹 ‘ 𝑧 ) ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ) → ( ( ◡ 𝐹 ‘ 𝑧 ) E 𝑥 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) E ( 𝐹 ‘ 𝑥 ) ) ) |
| 40 |
34 38 26 39
|
syl12anc |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ( ( ◡ 𝐹 ‘ 𝑧 ) E 𝑥 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) E ( 𝐹 ‘ 𝑥 ) ) ) |
| 41 |
|
epel |
⊢ ( ( ◡ 𝐹 ‘ 𝑧 ) E 𝑥 ↔ ( ◡ 𝐹 ‘ 𝑧 ) ∈ 𝑥 ) |
| 42 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑥 ) ∈ V |
| 43 |
42
|
epeli |
⊢ ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) E ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) ∈ ( 𝐹 ‘ 𝑥 ) ) |
| 44 |
40 41 43
|
3bitr3g |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ( ( ◡ 𝐹 ‘ 𝑧 ) ∈ 𝑥 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
| 45 |
33 44
|
mpbird |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ( ◡ 𝐹 ‘ 𝑧 ) ∈ 𝑥 ) |
| 46 |
|
simplrr |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) |
| 47 |
|
fveq2 |
⊢ ( 𝑦 = ( ◡ 𝐹 ‘ 𝑧 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) ) |
| 48 |
|
id |
⊢ ( 𝑦 = ( ◡ 𝐹 ‘ 𝑧 ) → 𝑦 = ( ◡ 𝐹 ‘ 𝑧 ) ) |
| 49 |
47 48
|
eqeq12d |
⊢ ( 𝑦 = ( ◡ 𝐹 ‘ 𝑧 ) → ( ( 𝐹 ‘ 𝑦 ) = 𝑦 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) = ( ◡ 𝐹 ‘ 𝑧 ) ) ) |
| 50 |
49
|
rspcv |
⊢ ( ( ◡ 𝐹 ‘ 𝑧 ) ∈ 𝑥 → ( ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) = ( ◡ 𝐹 ‘ 𝑧 ) ) ) |
| 51 |
45 46 50
|
sylc |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑧 ) ) = ( ◡ 𝐹 ‘ 𝑧 ) ) |
| 52 |
32 51
|
eqtr3d |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → 𝑧 = ( ◡ 𝐹 ‘ 𝑧 ) ) |
| 53 |
52 45
|
eqeltrd |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) → 𝑧 ∈ 𝑥 ) |
| 54 |
|
simprr |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) → ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) |
| 55 |
|
fveq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) |
| 56 |
|
id |
⊢ ( 𝑦 = 𝑧 → 𝑦 = 𝑧 ) |
| 57 |
55 56
|
eqeq12d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝐹 ‘ 𝑦 ) = 𝑦 ↔ ( 𝐹 ‘ 𝑧 ) = 𝑧 ) ) |
| 58 |
57
|
rspccva |
⊢ ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ∧ 𝑧 ∈ 𝑥 ) → ( 𝐹 ‘ 𝑧 ) = 𝑧 ) |
| 59 |
54 58
|
sylan |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ 𝑥 ) → ( 𝐹 ‘ 𝑧 ) = 𝑧 ) |
| 60 |
|
epel |
⊢ ( 𝑧 E 𝑥 ↔ 𝑧 ∈ 𝑥 ) |
| 61 |
60
|
biimpri |
⊢ ( 𝑧 ∈ 𝑥 → 𝑧 E 𝑥 ) |
| 62 |
61
|
adantl |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ 𝑥 ) → 𝑧 E 𝑥 ) |
| 63 |
|
simpll1 |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ 𝑥 ) → 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ) |
| 64 |
|
simpl2 |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) → Ord 𝐴 ) |
| 65 |
|
simprl |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) → 𝑥 ∈ 𝐴 ) |
| 66 |
64 65 11
|
syl2anc |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) → 𝑥 ⊆ 𝐴 ) |
| 67 |
66
|
sselda |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ 𝑥 ) → 𝑧 ∈ 𝐴 ) |
| 68 |
|
simplrl |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ 𝑥 ) → 𝑥 ∈ 𝐴 ) |
| 69 |
|
isorel |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ ( 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ) → ( 𝑧 E 𝑥 ↔ ( 𝐹 ‘ 𝑧 ) E ( 𝐹 ‘ 𝑥 ) ) ) |
| 70 |
63 67 68 69
|
syl12anc |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ 𝑥 ) → ( 𝑧 E 𝑥 ↔ ( 𝐹 ‘ 𝑧 ) E ( 𝐹 ‘ 𝑥 ) ) ) |
| 71 |
62 70
|
mpbid |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ 𝑥 ) → ( 𝐹 ‘ 𝑧 ) E ( 𝐹 ‘ 𝑥 ) ) |
| 72 |
42
|
epeli |
⊢ ( ( 𝐹 ‘ 𝑧 ) E ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑧 ) ∈ ( 𝐹 ‘ 𝑥 ) ) |
| 73 |
71 72
|
sylib |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ 𝑥 ) → ( 𝐹 ‘ 𝑧 ) ∈ ( 𝐹 ‘ 𝑥 ) ) |
| 74 |
59 73
|
eqeltrrd |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) ∧ 𝑧 ∈ 𝑥 ) → 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ) |
| 75 |
53 74
|
impbida |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) → ( 𝑧 ∈ ( 𝐹 ‘ 𝑥 ) ↔ 𝑧 ∈ 𝑥 ) ) |
| 76 |
75
|
eqrdv |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) |
| 77 |
76
|
expr |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = 𝑦 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) |
| 78 |
16 77
|
sylbid |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) |
| 79 |
78
|
ex |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) |
| 80 |
79
|
com23 |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) |
| 81 |
80
|
a2i |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) → ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) |
| 82 |
81
|
a1i |
⊢ ( 𝑥 ∈ On → ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) → ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) ) |
| 83 |
10 82
|
syl5bi |
⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑦 ∈ 𝐴 → ( 𝐹 ‘ 𝑦 ) = 𝑦 ) ) → ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) ) |
| 84 |
9 83
|
tfis2 |
⊢ ( 𝑥 ∈ On → ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) |
| 85 |
84
|
com3l |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ On → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) |
| 86 |
3 85
|
mpdd |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) |
| 87 |
86
|
ralrimiv |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) |
| 88 |
|
fveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑧 ) ) |
| 89 |
|
id |
⊢ ( 𝑥 = 𝑧 → 𝑥 = 𝑧 ) |
| 90 |
88 89
|
eqeq12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝐹 ‘ 𝑥 ) = 𝑥 ↔ ( 𝐹 ‘ 𝑧 ) = 𝑧 ) ) |
| 91 |
90
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) = 𝑧 ) |
| 92 |
91
|
adantll |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) = 𝑧 ) |
| 93 |
23
|
ffvelrnda |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝐵 ) |
| 94 |
93
|
3ad2antl1 |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝐵 ) |
| 95 |
94
|
adantlr |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝐵 ) |
| 96 |
92 95
|
eqeltrrd |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ 𝐵 ) |
| 97 |
96
|
ex |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → ( 𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵 ) ) |
| 98 |
|
simpl1 |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ) |
| 99 |
|
f1ofo |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 : 𝐴 –onto→ 𝐵 ) |
| 100 |
|
forn |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ran 𝐹 = 𝐵 ) |
| 101 |
17 99 100
|
3syl |
⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ran 𝐹 = 𝐵 ) |
| 102 |
98 101
|
syl |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → ran 𝐹 = 𝐵 ) |
| 103 |
102
|
eleq2d |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → ( 𝑧 ∈ ran 𝐹 ↔ 𝑧 ∈ 𝐵 ) ) |
| 104 |
|
f1ofn |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 Fn 𝐴 ) |
| 105 |
17 104
|
syl |
⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → 𝐹 Fn 𝐴 ) |
| 106 |
105
|
3ad2ant1 |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → 𝐹 Fn 𝐴 ) |
| 107 |
106
|
adantr |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → 𝐹 Fn 𝐴 ) |
| 108 |
|
fvelrnb |
⊢ ( 𝐹 Fn 𝐴 → ( 𝑧 ∈ ran 𝐹 ↔ ∃ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) = 𝑧 ) ) |
| 109 |
107 108
|
syl |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → ( 𝑧 ∈ ran 𝐹 ↔ ∃ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) = 𝑧 ) ) |
| 110 |
103 109
|
bitr3d |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → ( 𝑧 ∈ 𝐵 ↔ ∃ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) = 𝑧 ) ) |
| 111 |
|
fveq2 |
⊢ ( 𝑥 = 𝑤 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑤 ) ) |
| 112 |
|
id |
⊢ ( 𝑥 = 𝑤 → 𝑥 = 𝑤 ) |
| 113 |
111 112
|
eqeq12d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝐹 ‘ 𝑥 ) = 𝑥 ↔ ( 𝐹 ‘ 𝑤 ) = 𝑤 ) ) |
| 114 |
113
|
rspcv |
⊢ ( 𝑤 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 → ( 𝐹 ‘ 𝑤 ) = 𝑤 ) ) |
| 115 |
114
|
a1i |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑤 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 → ( 𝐹 ‘ 𝑤 ) = 𝑤 ) ) ) |
| 116 |
|
simpr |
⊢ ( ( ( 𝐹 ‘ 𝑤 ) = 𝑤 ∧ ( 𝐹 ‘ 𝑤 ) = 𝑧 ) → ( 𝐹 ‘ 𝑤 ) = 𝑧 ) |
| 117 |
|
simpl |
⊢ ( ( ( 𝐹 ‘ 𝑤 ) = 𝑤 ∧ ( 𝐹 ‘ 𝑤 ) = 𝑧 ) → ( 𝐹 ‘ 𝑤 ) = 𝑤 ) |
| 118 |
116 117
|
eqtr3d |
⊢ ( ( ( 𝐹 ‘ 𝑤 ) = 𝑤 ∧ ( 𝐹 ‘ 𝑤 ) = 𝑧 ) → 𝑧 = 𝑤 ) |
| 119 |
118
|
adantl |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑤 ) = 𝑤 ∧ ( 𝐹 ‘ 𝑤 ) = 𝑧 ) ) → 𝑧 = 𝑤 ) |
| 120 |
|
simplr |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑤 ) = 𝑤 ∧ ( 𝐹 ‘ 𝑤 ) = 𝑧 ) ) → 𝑤 ∈ 𝐴 ) |
| 121 |
119 120
|
eqeltrd |
⊢ ( ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑤 ) = 𝑤 ∧ ( 𝐹 ‘ 𝑤 ) = 𝑧 ) ) → 𝑧 ∈ 𝐴 ) |
| 122 |
121
|
exp43 |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑤 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑤 ) = 𝑤 → ( ( 𝐹 ‘ 𝑤 ) = 𝑧 → 𝑧 ∈ 𝐴 ) ) ) ) |
| 123 |
115 122
|
syldd |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝑤 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 → ( ( 𝐹 ‘ 𝑤 ) = 𝑧 → 𝑧 ∈ 𝐴 ) ) ) ) |
| 124 |
123
|
com23 |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 → ( 𝑤 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑤 ) = 𝑧 → 𝑧 ∈ 𝐴 ) ) ) ) |
| 125 |
124
|
imp |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → ( 𝑤 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑤 ) = 𝑧 → 𝑧 ∈ 𝐴 ) ) ) |
| 126 |
125
|
rexlimdv |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → ( ∃ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) = 𝑧 → 𝑧 ∈ 𝐴 ) ) |
| 127 |
110 126
|
sylbid |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐴 ) ) |
| 128 |
97 127
|
impbid |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → ( 𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵 ) ) |
| 129 |
128
|
eqrdv |
⊢ ( ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑥 ) → 𝐴 = 𝐵 ) |
| 130 |
87 129
|
mpdan |
⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ Ord 𝐵 ) → 𝐴 = 𝐵 ) |