Step |
Hyp |
Ref |
Expression |
1 |
|
sdomdom |
⊢ ( 𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵 ) |
2 |
|
relsdom |
⊢ Rel ≺ |
3 |
2
|
brrelex2i |
⊢ ( 𝐴 ≺ 𝐵 → 𝐵 ∈ V ) |
4 |
|
brdomg |
⊢ ( 𝐵 ∈ V → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) ) |
5 |
3 4
|
syl |
⊢ ( 𝐴 ≺ 𝐵 → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) ) |
6 |
1 5
|
mpbid |
⊢ ( 𝐴 ≺ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) |
7 |
6
|
adantr |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) |
8 |
|
f1f |
⊢ ( 𝑓 : 𝐴 –1-1→ 𝐵 → 𝑓 : 𝐴 ⟶ 𝐵 ) |
9 |
8
|
frnd |
⊢ ( 𝑓 : 𝐴 –1-1→ 𝐵 → ran 𝑓 ⊆ 𝐵 ) |
10 |
9
|
adantl |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ran 𝑓 ⊆ 𝐵 ) |
11 |
|
sdomnen |
⊢ ( 𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵 ) |
12 |
11
|
ad2antrr |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ¬ 𝐴 ≈ 𝐵 ) |
13 |
|
vex |
⊢ 𝑓 ∈ V |
14 |
|
dff1o5 |
⊢ ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ ran 𝑓 = 𝐵 ) ) |
15 |
14
|
biimpri |
⊢ ( ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ ran 𝑓 = 𝐵 ) → 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) |
16 |
|
f1oen3g |
⊢ ( ( 𝑓 ∈ V ∧ 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 ) |
17 |
13 15 16
|
sylancr |
⊢ ( ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ ran 𝑓 = 𝐵 ) → 𝐴 ≈ 𝐵 ) |
18 |
17
|
ex |
⊢ ( 𝑓 : 𝐴 –1-1→ 𝐵 → ( ran 𝑓 = 𝐵 → 𝐴 ≈ 𝐵 ) ) |
19 |
18
|
necon3bd |
⊢ ( 𝑓 : 𝐴 –1-1→ 𝐵 → ( ¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵 ) ) |
20 |
19
|
adantl |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( ¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵 ) ) |
21 |
12 20
|
mpd |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ran 𝑓 ≠ 𝐵 ) |
22 |
|
pssdifn0 |
⊢ ( ( ran 𝑓 ⊆ 𝐵 ∧ ran 𝑓 ≠ 𝐵 ) → ( 𝐵 ∖ ran 𝑓 ) ≠ ∅ ) |
23 |
10 21 22
|
syl2anc |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( 𝐵 ∖ ran 𝑓 ) ≠ ∅ ) |
24 |
|
n0 |
⊢ ( ( 𝐵 ∖ ran 𝑓 ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) |
25 |
23 24
|
sylib |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ∃ 𝑥 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) |
26 |
2
|
brrelex1i |
⊢ ( 𝐴 ≺ 𝐵 → 𝐴 ∈ V ) |
27 |
26
|
ad2antrr |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) ) → 𝐴 ∈ V ) |
28 |
3
|
ad2antrr |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) ) → 𝐵 ∈ V ) |
29 |
28
|
difexd |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) ) → ( 𝐵 ∖ { 𝑥 } ) ∈ V ) |
30 |
|
eldifn |
⊢ ( 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) → ¬ 𝑥 ∈ ran 𝑓 ) |
31 |
|
disjsn |
⊢ ( ( ran 𝑓 ∩ { 𝑥 } ) = ∅ ↔ ¬ 𝑥 ∈ ran 𝑓 ) |
32 |
30 31
|
sylibr |
⊢ ( 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) → ( ran 𝑓 ∩ { 𝑥 } ) = ∅ ) |
33 |
32
|
adantl |
⊢ ( ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) → ( ran 𝑓 ∩ { 𝑥 } ) = ∅ ) |
34 |
9
|
adantr |
⊢ ( ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) → ran 𝑓 ⊆ 𝐵 ) |
35 |
|
reldisj |
⊢ ( ran 𝑓 ⊆ 𝐵 → ( ( ran 𝑓 ∩ { 𝑥 } ) = ∅ ↔ ran 𝑓 ⊆ ( 𝐵 ∖ { 𝑥 } ) ) ) |
36 |
34 35
|
syl |
⊢ ( ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) → ( ( ran 𝑓 ∩ { 𝑥 } ) = ∅ ↔ ran 𝑓 ⊆ ( 𝐵 ∖ { 𝑥 } ) ) ) |
37 |
33 36
|
mpbid |
⊢ ( ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) → ran 𝑓 ⊆ ( 𝐵 ∖ { 𝑥 } ) ) |
38 |
|
f1ssr |
⊢ ( ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ ran 𝑓 ⊆ ( 𝐵 ∖ { 𝑥 } ) ) → 𝑓 : 𝐴 –1-1→ ( 𝐵 ∖ { 𝑥 } ) ) |
39 |
37 38
|
syldan |
⊢ ( ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) → 𝑓 : 𝐴 –1-1→ ( 𝐵 ∖ { 𝑥 } ) ) |
40 |
39
|
adantl |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) ) → 𝑓 : 𝐴 –1-1→ ( 𝐵 ∖ { 𝑥 } ) ) |
41 |
|
f1dom2g |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝐵 ∖ { 𝑥 } ) ∈ V ∧ 𝑓 : 𝐴 –1-1→ ( 𝐵 ∖ { 𝑥 } ) ) → 𝐴 ≼ ( 𝐵 ∖ { 𝑥 } ) ) |
42 |
27 29 40 41
|
syl3anc |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) ) → 𝐴 ≼ ( 𝐵 ∖ { 𝑥 } ) ) |
43 |
|
eldifi |
⊢ ( 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) → 𝑥 ∈ 𝐵 ) |
44 |
43
|
ad2antll |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) ) → 𝑥 ∈ 𝐵 ) |
45 |
|
simplr |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) ) → 𝐶 ∈ 𝐵 ) |
46 |
|
difsnen |
⊢ ( ( 𝐵 ∈ V ∧ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → ( 𝐵 ∖ { 𝑥 } ) ≈ ( 𝐵 ∖ { 𝐶 } ) ) |
47 |
28 44 45 46
|
syl3anc |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) ) → ( 𝐵 ∖ { 𝑥 } ) ≈ ( 𝐵 ∖ { 𝐶 } ) ) |
48 |
|
domentr |
⊢ ( ( 𝐴 ≼ ( 𝐵 ∖ { 𝑥 } ) ∧ ( 𝐵 ∖ { 𝑥 } ) ≈ ( 𝐵 ∖ { 𝐶 } ) ) → 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ) |
49 |
42 47 48
|
syl2anc |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑓 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) ) ) → 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ) |
50 |
49
|
expr |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) → 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ) ) |
51 |
50
|
exlimdv |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → ( ∃ 𝑥 𝑥 ∈ ( 𝐵 ∖ ran 𝑓 ) → 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ) ) |
52 |
25 51
|
mpd |
⊢ ( ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑓 : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ) |
53 |
7 52
|
exlimddv |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ) |
54 |
1
|
adantr |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵 ) → 𝐴 ≼ 𝐵 ) |
55 |
|
difsn |
⊢ ( ¬ 𝐶 ∈ 𝐵 → ( 𝐵 ∖ { 𝐶 } ) = 𝐵 ) |
56 |
55
|
breq2d |
⊢ ( ¬ 𝐶 ∈ 𝐵 → ( 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ↔ 𝐴 ≼ 𝐵 ) ) |
57 |
56
|
adantl |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵 ) → ( 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ↔ 𝐴 ≼ 𝐵 ) ) |
58 |
54 57
|
mpbird |
⊢ ( ( 𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵 ) → 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ) |
59 |
53 58
|
pm2.61dan |
⊢ ( 𝐴 ≺ 𝐵 → 𝐴 ≼ ( 𝐵 ∖ { 𝐶 } ) ) |