Step |
Hyp |
Ref |
Expression |
1 |
|
sdomdom |
⊢ ( 𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵 ) |
2 |
|
brdomi |
⊢ ( 𝐴 ≼ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) |
3 |
1 2
|
syl |
⊢ ( 𝐴 ≺ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) |
4 |
|
vex |
⊢ 𝑓 ∈ V |
5 |
4
|
rnex |
⊢ ran 𝑓 ∈ V |
6 |
|
f1f1orn |
⊢ ( 𝑓 : 𝐴 –1-1→ 𝐵 → 𝑓 : 𝐴 –1-1-onto→ ran 𝑓 ) |
7 |
6
|
adantl |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → 𝑓 : 𝐴 –1-1-onto→ ran 𝑓 ) |
8 |
|
f1of1 |
⊢ ( 𝑓 : 𝐴 –1-1-onto→ ran 𝑓 → 𝑓 : 𝐴 –1-1→ ran 𝑓 ) |
9 |
7 8
|
syl |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → 𝑓 : 𝐴 –1-1→ ran 𝑓 ) |
10 |
|
f1dom3g |
⊢ ( ( 𝑓 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓 : 𝐴 –1-1→ ran 𝑓 ) → 𝐴 ≼ ran 𝑓 ) |
11 |
4 5 9 10
|
mp3an12i |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ ran 𝑓 ) |
12 |
|
sdomnen |
⊢ ( 𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵 ) |
13 |
12
|
adantr |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ¬ 𝐴 ≈ 𝐵 ) |
14 |
|
ssdif0 |
⊢ ( 𝐵 ⊆ ran 𝑓 ↔ ( 𝐵 ∖ ran 𝑓 ) = ∅ ) |
15 |
|
simplr |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) ∧ 𝐵 ⊆ ran 𝑓 ) → 𝑓 : 𝐴 –1-1→ 𝐵 ) |
16 |
|
f1f |
⊢ ( 𝑓 : 𝐴 –1-1→ 𝐵 → 𝑓 : 𝐴 ⟶ 𝐵 ) |
17 |
16
|
frnd |
⊢ ( 𝑓 : 𝐴 –1-1→ 𝐵 → ran 𝑓 ⊆ 𝐵 ) |
18 |
15 17
|
syl |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) ∧ 𝐵 ⊆ ran 𝑓 ) → ran 𝑓 ⊆ 𝐵 ) |
19 |
|
simpr |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) ∧ 𝐵 ⊆ ran 𝑓 ) → 𝐵 ⊆ ran 𝑓 ) |
20 |
18 19
|
eqssd |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) ∧ 𝐵 ⊆ ran 𝑓 ) → ran 𝑓 = 𝐵 ) |
21 |
|
dff1o5 |
⊢ ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ ran 𝑓 = 𝐵 ) ) |
22 |
15 20 21
|
sylanbrc |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) ∧ 𝐵 ⊆ ran 𝑓 ) → 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) |
23 |
|
f1oen3g |
⊢ ( ( 𝑓 ∈ V ∧ 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 ) |
24 |
4 22 23
|
sylancr |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) ∧ 𝐵 ⊆ ran 𝑓 ) → 𝐴 ≈ 𝐵 ) |
25 |
24
|
ex |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( 𝐵 ⊆ ran 𝑓 → 𝐴 ≈ 𝐵 ) ) |
26 |
14 25
|
biimtrrid |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( ( 𝐵 ∖ ran 𝑓 ) = ∅ → 𝐴 ≈ 𝐵 ) ) |
27 |
13 26
|
mtod |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ¬ ( 𝐵 ∖ ran 𝑓 ) = ∅ ) |
28 |
|
neq0 |
⊢ ( ¬ ( 𝐵 ∖ ran 𝑓 ) = ∅ ↔ ∃ 𝑤 𝑤 ∈ ( 𝐵 ∖ ran 𝑓 ) ) |
29 |
27 28
|
sylib |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ∃ 𝑤 𝑤 ∈ ( 𝐵 ∖ ran 𝑓 ) ) |
30 |
|
snssi |
⊢ ( 𝑤 ∈ ( 𝐵 ∖ ran 𝑓 ) → { 𝑤 } ⊆ ( 𝐵 ∖ ran 𝑓 ) ) |
31 |
|
relsdom |
⊢ Rel ≺ |
32 |
31
|
brrelex1i |
⊢ ( 𝐴 ≺ 𝐵 → 𝐴 ∈ V ) |
33 |
32
|
adantr |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → 𝐴 ∈ V ) |
34 |
|
vex |
⊢ 𝑤 ∈ V |
35 |
|
en2sn |
⊢ ( ( 𝐴 ∈ V ∧ 𝑤 ∈ V ) → { 𝐴 } ≈ { 𝑤 } ) |
36 |
33 34 35
|
sylancl |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → { 𝐴 } ≈ { 𝑤 } ) |
37 |
31
|
brrelex2i |
⊢ ( 𝐴 ≺ 𝐵 → 𝐵 ∈ V ) |
38 |
37
|
adantr |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → 𝐵 ∈ V ) |
39 |
|
difexg |
⊢ ( 𝐵 ∈ V → ( 𝐵 ∖ ran 𝑓 ) ∈ V ) |
40 |
|
snfi |
⊢ { 𝑤 } ∈ Fin |
41 |
|
ssdomfi2 |
⊢ ( ( { 𝑤 } ∈ Fin ∧ ( 𝐵 ∖ ran 𝑓 ) ∈ V ∧ { 𝑤 } ⊆ ( 𝐵 ∖ ran 𝑓 ) ) → { 𝑤 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) |
42 |
40 41
|
mp3an1 |
⊢ ( ( ( 𝐵 ∖ ran 𝑓 ) ∈ V ∧ { 𝑤 } ⊆ ( 𝐵 ∖ ran 𝑓 ) ) → { 𝑤 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) |
43 |
42
|
ex |
⊢ ( ( 𝐵 ∖ ran 𝑓 ) ∈ V → ( { 𝑤 } ⊆ ( 𝐵 ∖ ran 𝑓 ) → { 𝑤 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) ) |
44 |
38 39 43
|
3syl |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( { 𝑤 } ⊆ ( 𝐵 ∖ ran 𝑓 ) → { 𝑤 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) ) |
45 |
|
endom |
⊢ ( { 𝐴 } ≈ { 𝑤 } → { 𝐴 } ≼ { 𝑤 } ) |
46 |
|
domtrfi |
⊢ ( ( { 𝑤 } ∈ Fin ∧ { 𝐴 } ≼ { 𝑤 } ∧ { 𝑤 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) → { 𝐴 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) |
47 |
40 46
|
mp3an1 |
⊢ ( ( { 𝐴 } ≼ { 𝑤 } ∧ { 𝑤 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) → { 𝐴 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) |
48 |
45 47
|
sylan |
⊢ ( ( { 𝐴 } ≈ { 𝑤 } ∧ { 𝑤 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) → { 𝐴 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) |
49 |
36 44 48
|
syl6an |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( { 𝑤 } ⊆ ( 𝐵 ∖ ran 𝑓 ) → { 𝐴 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) ) |
50 |
30 49
|
syl5 |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( 𝑤 ∈ ( 𝐵 ∖ ran 𝑓 ) → { 𝐴 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) ) |
51 |
50
|
exlimdv |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( ∃ 𝑤 𝑤 ∈ ( 𝐵 ∖ ran 𝑓 ) → { 𝐴 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) ) |
52 |
29 51
|
mpd |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → { 𝐴 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) |
53 |
|
disjdif |
⊢ ( ran 𝑓 ∩ ( 𝐵 ∖ ran 𝑓 ) ) = ∅ |
54 |
53
|
a1i |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( ran 𝑓 ∩ ( 𝐵 ∖ ran 𝑓 ) ) = ∅ ) |
55 |
|
undom |
⊢ ( ( ( 𝐴 ≼ ran 𝑓 ∧ { 𝐴 } ≼ ( 𝐵 ∖ ran 𝑓 ) ) ∧ ( ran 𝑓 ∩ ( 𝐵 ∖ ran 𝑓 ) ) = ∅ ) → ( 𝐴 ∪ { 𝐴 } ) ≼ ( ran 𝑓 ∪ ( 𝐵 ∖ ran 𝑓 ) ) ) |
56 |
11 52 54 55
|
syl21anc |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( 𝐴 ∪ { 𝐴 } ) ≼ ( ran 𝑓 ∪ ( 𝐵 ∖ ran 𝑓 ) ) ) |
57 |
|
df-suc |
⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } ) |
58 |
57
|
a1i |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → suc 𝐴 = ( 𝐴 ∪ { 𝐴 } ) ) |
59 |
|
undif2 |
⊢ ( ran 𝑓 ∪ ( 𝐵 ∖ ran 𝑓 ) ) = ( ran 𝑓 ∪ 𝐵 ) |
60 |
17
|
adantl |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ran 𝑓 ⊆ 𝐵 ) |
61 |
|
ssequn1 |
⊢ ( ran 𝑓 ⊆ 𝐵 ↔ ( ran 𝑓 ∪ 𝐵 ) = 𝐵 ) |
62 |
60 61
|
sylib |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( ran 𝑓 ∪ 𝐵 ) = 𝐵 ) |
63 |
59 62
|
eqtr2id |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → 𝐵 = ( ran 𝑓 ∪ ( 𝐵 ∖ ran 𝑓 ) ) ) |
64 |
56 58 63
|
3brtr4d |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → suc 𝐴 ≼ 𝐵 ) |
65 |
3 64
|
exlimddv |
⊢ ( 𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵 ) |