Step |
Hyp |
Ref |
Expression |
1 |
|
phplem2.1 |
⊢ 𝐴 ∈ V |
2 |
|
bren |
⊢ ( suc 𝐴 ≈ suc 𝐵 ↔ ∃ 𝑓 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) |
3 |
|
f1of1 |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → 𝑓 : suc 𝐴 –1-1→ suc 𝐵 ) |
4 |
|
nnfi |
⊢ ( 𝐴 ∈ ω → 𝐴 ∈ Fin ) |
5 |
|
sssucid |
⊢ 𝐴 ⊆ suc 𝐴 |
6 |
|
f1imaenfi |
⊢ ( ( 𝑓 : suc 𝐴 –1-1→ suc 𝐵 ∧ 𝐴 ⊆ suc 𝐴 ∧ 𝐴 ∈ Fin ) → ( 𝑓 “ 𝐴 ) ≈ 𝐴 ) |
7 |
5 6
|
mp3an2 |
⊢ ( ( 𝑓 : suc 𝐴 –1-1→ suc 𝐵 ∧ 𝐴 ∈ Fin ) → ( 𝑓 “ 𝐴 ) ≈ 𝐴 ) |
8 |
3 4 7
|
syl2anr |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) → ( 𝑓 “ 𝐴 ) ≈ 𝐴 ) |
9 |
|
ensymfib |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ≈ ( 𝑓 “ 𝐴 ) ↔ ( 𝑓 “ 𝐴 ) ≈ 𝐴 ) ) |
10 |
4 9
|
syl |
⊢ ( 𝐴 ∈ ω → ( 𝐴 ≈ ( 𝑓 “ 𝐴 ) ↔ ( 𝑓 “ 𝐴 ) ≈ 𝐴 ) ) |
11 |
10
|
adantr |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) → ( 𝐴 ≈ ( 𝑓 “ 𝐴 ) ↔ ( 𝑓 “ 𝐴 ) ≈ 𝐴 ) ) |
12 |
8 11
|
mpbird |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) → 𝐴 ≈ ( 𝑓 “ 𝐴 ) ) |
13 |
|
nnord |
⊢ ( 𝐴 ∈ ω → Ord 𝐴 ) |
14 |
|
orddif |
⊢ ( Ord 𝐴 → 𝐴 = ( suc 𝐴 ∖ { 𝐴 } ) ) |
15 |
13 14
|
syl |
⊢ ( 𝐴 ∈ ω → 𝐴 = ( suc 𝐴 ∖ { 𝐴 } ) ) |
16 |
15
|
imaeq2d |
⊢ ( 𝐴 ∈ ω → ( 𝑓 “ 𝐴 ) = ( 𝑓 “ ( suc 𝐴 ∖ { 𝐴 } ) ) ) |
17 |
|
f1ofn |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → 𝑓 Fn suc 𝐴 ) |
18 |
1
|
sucid |
⊢ 𝐴 ∈ suc 𝐴 |
19 |
|
fnsnfv |
⊢ ( ( 𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴 ) → { ( 𝑓 ‘ 𝐴 ) } = ( 𝑓 “ { 𝐴 } ) ) |
20 |
17 18 19
|
sylancl |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → { ( 𝑓 ‘ 𝐴 ) } = ( 𝑓 “ { 𝐴 } ) ) |
21 |
20
|
difeq2d |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → ( ( 𝑓 “ suc 𝐴 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) = ( ( 𝑓 “ suc 𝐴 ) ∖ ( 𝑓 “ { 𝐴 } ) ) ) |
22 |
|
imadmrn |
⊢ ( 𝑓 “ dom 𝑓 ) = ran 𝑓 |
23 |
22
|
eqcomi |
⊢ ran 𝑓 = ( 𝑓 “ dom 𝑓 ) |
24 |
|
f1ofo |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → 𝑓 : suc 𝐴 –onto→ suc 𝐵 ) |
25 |
|
forn |
⊢ ( 𝑓 : suc 𝐴 –onto→ suc 𝐵 → ran 𝑓 = suc 𝐵 ) |
26 |
24 25
|
syl |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → ran 𝑓 = suc 𝐵 ) |
27 |
|
f1odm |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → dom 𝑓 = suc 𝐴 ) |
28 |
27
|
imaeq2d |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → ( 𝑓 “ dom 𝑓 ) = ( 𝑓 “ suc 𝐴 ) ) |
29 |
23 26 28
|
3eqtr3a |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → suc 𝐵 = ( 𝑓 “ suc 𝐴 ) ) |
30 |
29
|
difeq1d |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) = ( ( 𝑓 “ suc 𝐴 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) |
31 |
|
dff1o3 |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ↔ ( 𝑓 : suc 𝐴 –onto→ suc 𝐵 ∧ Fun ◡ 𝑓 ) ) |
32 |
|
imadif |
⊢ ( Fun ◡ 𝑓 → ( 𝑓 “ ( suc 𝐴 ∖ { 𝐴 } ) ) = ( ( 𝑓 “ suc 𝐴 ) ∖ ( 𝑓 “ { 𝐴 } ) ) ) |
33 |
31 32
|
simplbiim |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → ( 𝑓 “ ( suc 𝐴 ∖ { 𝐴 } ) ) = ( ( 𝑓 “ suc 𝐴 ) ∖ ( 𝑓 “ { 𝐴 } ) ) ) |
34 |
21 30 33
|
3eqtr4rd |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → ( 𝑓 “ ( suc 𝐴 ∖ { 𝐴 } ) ) = ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) |
35 |
16 34
|
sylan9eq |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) → ( 𝑓 “ 𝐴 ) = ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) |
36 |
12 35
|
breqtrd |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) → 𝐴 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) |
37 |
|
fnfvelrn |
⊢ ( ( 𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴 ) → ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑓 ) |
38 |
17 18 37
|
sylancl |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑓 ) |
39 |
25
|
eleq2d |
⊢ ( 𝑓 : suc 𝐴 –onto→ suc 𝐵 → ( ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑓 ↔ ( 𝑓 ‘ 𝐴 ) ∈ suc 𝐵 ) ) |
40 |
24 39
|
syl |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → ( ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑓 ↔ ( 𝑓 ‘ 𝐴 ) ∈ suc 𝐵 ) ) |
41 |
38 40
|
mpbid |
⊢ ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → ( 𝑓 ‘ 𝐴 ) ∈ suc 𝐵 ) |
42 |
|
phplem1 |
⊢ ( ( 𝐵 ∈ ω ∧ ( 𝑓 ‘ 𝐴 ) ∈ suc 𝐵 ) → 𝐵 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) |
43 |
41 42
|
sylan2 |
⊢ ( ( 𝐵 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) → 𝐵 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) |
44 |
|
nnfi |
⊢ ( 𝐵 ∈ ω → 𝐵 ∈ Fin ) |
45 |
|
ensymfib |
⊢ ( 𝐵 ∈ Fin → ( 𝐵 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ↔ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ≈ 𝐵 ) ) |
46 |
44 45
|
syl |
⊢ ( 𝐵 ∈ ω → ( 𝐵 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ↔ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ≈ 𝐵 ) ) |
47 |
46
|
adantr |
⊢ ( ( 𝐵 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) → ( 𝐵 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ↔ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ≈ 𝐵 ) ) |
48 |
43 47
|
mpbid |
⊢ ( ( 𝐵 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) → ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ≈ 𝐵 ) |
49 |
|
entrfil |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐴 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ∧ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ≈ 𝐵 ) → 𝐴 ≈ 𝐵 ) |
50 |
4 49
|
syl3an1 |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐴 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ∧ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ≈ 𝐵 ) → 𝐴 ≈ 𝐵 ) |
51 |
48 50
|
syl3an3 |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐴 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ∧ ( 𝐵 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) ) → 𝐴 ≈ 𝐵 ) |
52 |
51
|
3expa |
⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐴 ≈ ( suc 𝐵 ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) ∧ ( 𝐵 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) ) → 𝐴 ≈ 𝐵 ) |
53 |
36 52
|
syldanl |
⊢ ( ( ( 𝐴 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) ∧ ( 𝐵 ∈ ω ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) ) → 𝐴 ≈ 𝐵 ) |
54 |
53
|
anandirs |
⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 ) → 𝐴 ≈ 𝐵 ) |
55 |
54
|
ex |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → 𝐴 ≈ 𝐵 ) ) |
56 |
55
|
exlimdv |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ∃ 𝑓 𝑓 : suc 𝐴 –1-1-onto→ suc 𝐵 → 𝐴 ≈ 𝐵 ) ) |
57 |
2 56
|
syl5bi |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵 ) ) |