Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) → 𝐹 : 𝐴 –onto→ 𝐵 ) |
2 |
|
fof |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
3 |
1 2
|
syl |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
4 |
|
domnsym |
⊢ ( 𝐵 ≼ ( 𝐴 ∖ { 𝑦 } ) → ¬ ( 𝐴 ∖ { 𝑦 } ) ≺ 𝐵 ) |
5 |
|
simp3 |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) → 𝐵 ∈ Fin ) |
6 |
|
simp2 |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) → 𝐴 ≈ 𝐵 ) |
7 |
|
enfii |
⊢ ( ( 𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵 ) → 𝐴 ∈ Fin ) |
8 |
5 6 7
|
syl2anc |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) → 𝐴 ∈ Fin ) |
9 |
8
|
ad2antrr |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → 𝐴 ∈ Fin ) |
10 |
|
difssd |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → ( 𝐴 ∖ { 𝑦 } ) ⊆ 𝐴 ) |
11 |
|
simplrr |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → 𝑦 ∈ 𝐴 ) |
12 |
|
neldifsn |
⊢ ¬ 𝑦 ∈ ( 𝐴 ∖ { 𝑦 } ) |
13 |
|
nelne1 |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ ( 𝐴 ∖ { 𝑦 } ) ) → 𝐴 ≠ ( 𝐴 ∖ { 𝑦 } ) ) |
14 |
11 12 13
|
sylancl |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → 𝐴 ≠ ( 𝐴 ∖ { 𝑦 } ) ) |
15 |
14
|
necomd |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → ( 𝐴 ∖ { 𝑦 } ) ≠ 𝐴 ) |
16 |
|
df-pss |
⊢ ( ( 𝐴 ∖ { 𝑦 } ) ⊊ 𝐴 ↔ ( ( 𝐴 ∖ { 𝑦 } ) ⊆ 𝐴 ∧ ( 𝐴 ∖ { 𝑦 } ) ≠ 𝐴 ) ) |
17 |
10 15 16
|
sylanbrc |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → ( 𝐴 ∖ { 𝑦 } ) ⊊ 𝐴 ) |
18 |
|
php3 |
⊢ ( ( 𝐴 ∈ Fin ∧ ( 𝐴 ∖ { 𝑦 } ) ⊊ 𝐴 ) → ( 𝐴 ∖ { 𝑦 } ) ≺ 𝐴 ) |
19 |
9 17 18
|
syl2anc |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → ( 𝐴 ∖ { 𝑦 } ) ≺ 𝐴 ) |
20 |
6
|
ad2antrr |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → 𝐴 ≈ 𝐵 ) |
21 |
|
sdomentr |
⊢ ( ( ( 𝐴 ∖ { 𝑦 } ) ≺ 𝐴 ∧ 𝐴 ≈ 𝐵 ) → ( 𝐴 ∖ { 𝑦 } ) ≺ 𝐵 ) |
22 |
19 20 21
|
syl2anc |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → ( 𝐴 ∖ { 𝑦 } ) ≺ 𝐵 ) |
23 |
4 22
|
nsyl3 |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → ¬ 𝐵 ≼ ( 𝐴 ∖ { 𝑦 } ) ) |
24 |
8
|
adantr |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → 𝐴 ∈ Fin ) |
25 |
|
difss |
⊢ ( 𝐴 ∖ { 𝑦 } ) ⊆ 𝐴 |
26 |
|
ssfi |
⊢ ( ( 𝐴 ∈ Fin ∧ ( 𝐴 ∖ { 𝑦 } ) ⊆ 𝐴 ) → ( 𝐴 ∖ { 𝑦 } ) ∈ Fin ) |
27 |
24 25 26
|
sylancl |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → ( 𝐴 ∖ { 𝑦 } ) ∈ Fin ) |
28 |
3
|
adantr |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
29 |
|
fssres |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐴 ∖ { 𝑦 } ) ⊆ 𝐴 ) → ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) : ( 𝐴 ∖ { 𝑦 } ) ⟶ 𝐵 ) |
30 |
28 25 29
|
sylancl |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) : ( 𝐴 ∖ { 𝑦 } ) ⟶ 𝐵 ) |
31 |
1
|
adantr |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → 𝐹 : 𝐴 –onto→ 𝐵 ) |
32 |
|
foelrn |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ∃ 𝑢 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑢 ) ) |
33 |
31 32
|
sylan |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) ∧ 𝑧 ∈ 𝐵 ) → ∃ 𝑢 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑢 ) ) |
34 |
|
simprll |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → 𝑥 ∈ 𝐴 ) |
35 |
|
simprrr |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → 𝑥 ≠ 𝑦 ) |
36 |
|
eldifsn |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ { 𝑦 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝑦 ) ) |
37 |
34 35 36
|
sylanbrc |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → 𝑥 ∈ ( 𝐴 ∖ { 𝑦 } ) ) |
38 |
|
simprrl |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) |
39 |
38
|
eqcomd |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑥 ) ) |
40 |
|
fveq2 |
⊢ ( 𝑤 = 𝑥 → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑥 ) ) |
41 |
40
|
rspceeqv |
⊢ ( ( 𝑥 ∈ ( 𝐴 ∖ { 𝑦 } ) ∧ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑥 ) ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑤 ) ) |
42 |
37 39 41
|
syl2anc |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑤 ) ) |
43 |
|
fveqeq2 |
⊢ ( 𝑢 = 𝑦 → ( ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ↔ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑤 ) ) ) |
44 |
43
|
rexbidv |
⊢ ( 𝑢 = 𝑦 → ( ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ↔ ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑤 ) ) ) |
45 |
42 44
|
syl5ibrcom |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → ( 𝑢 = 𝑦 → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) ) |
46 |
45
|
adantr |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) ∧ 𝑢 ∈ 𝐴 ) → ( 𝑢 = 𝑦 → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) ) |
47 |
46
|
imp |
⊢ ( ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 = 𝑦 ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) |
48 |
|
eldifsn |
⊢ ( 𝑢 ∈ ( 𝐴 ∖ { 𝑦 } ) ↔ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ≠ 𝑦 ) ) |
49 |
|
eqid |
⊢ ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑢 ) |
50 |
|
fveq2 |
⊢ ( 𝑤 = 𝑢 → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑢 ) ) |
51 |
50
|
rspceeqv |
⊢ ( ( 𝑢 ∈ ( 𝐴 ∖ { 𝑦 } ) ∧ ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑢 ) ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) |
52 |
49 51
|
mpan2 |
⊢ ( 𝑢 ∈ ( 𝐴 ∖ { 𝑦 } ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) |
53 |
48 52
|
sylbir |
⊢ ( ( 𝑢 ∈ 𝐴 ∧ 𝑢 ≠ 𝑦 ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) |
54 |
53
|
adantll |
⊢ ( ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≠ 𝑦 ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) |
55 |
47 54
|
pm2.61dane |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) ∧ 𝑢 ∈ 𝐴 ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) |
56 |
|
fvres |
⊢ ( 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) → ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) = ( 𝐹 ‘ 𝑤 ) ) |
57 |
56
|
eqeq2d |
⊢ ( 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) → ( 𝑧 = ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) ↔ 𝑧 = ( 𝐹 ‘ 𝑤 ) ) ) |
58 |
57
|
rexbiia |
⊢ ( ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) ↔ ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( 𝐹 ‘ 𝑤 ) ) |
59 |
|
eqeq1 |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑢 ) → ( 𝑧 = ( 𝐹 ‘ 𝑤 ) ↔ ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) ) |
60 |
59
|
rexbidv |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑢 ) → ( ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( 𝐹 ‘ 𝑤 ) ↔ ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) ) |
61 |
58 60
|
syl5bb |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑢 ) → ( ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) ↔ ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) ( 𝐹 ‘ 𝑢 ) = ( 𝐹 ‘ 𝑤 ) ) ) |
62 |
55 61
|
syl5ibrcom |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) ∧ 𝑢 ∈ 𝐴 ) → ( 𝑧 = ( 𝐹 ‘ 𝑢 ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) ) ) |
63 |
62
|
rexlimdva |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → ( ∃ 𝑢 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑢 ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) ) ) |
64 |
63
|
imp |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) ∧ ∃ 𝑢 ∈ 𝐴 𝑧 = ( 𝐹 ‘ 𝑢 ) ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) ) |
65 |
33 64
|
syldan |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) ∧ 𝑧 ∈ 𝐵 ) → ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) ) |
66 |
65
|
ralrimiva |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → ∀ 𝑧 ∈ 𝐵 ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) ) |
67 |
|
dffo3 |
⊢ ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) : ( 𝐴 ∖ { 𝑦 } ) –onto→ 𝐵 ↔ ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) : ( 𝐴 ∖ { 𝑦 } ) ⟶ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ∃ 𝑤 ∈ ( 𝐴 ∖ { 𝑦 } ) 𝑧 = ( ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) ‘ 𝑤 ) ) ) |
68 |
30 66 67
|
sylanbrc |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) : ( 𝐴 ∖ { 𝑦 } ) –onto→ 𝐵 ) |
69 |
|
fodomfi |
⊢ ( ( ( 𝐴 ∖ { 𝑦 } ) ∈ Fin ∧ ( 𝐹 ↾ ( 𝐴 ∖ { 𝑦 } ) ) : ( 𝐴 ∖ { 𝑦 } ) –onto→ 𝐵 ) → 𝐵 ≼ ( 𝐴 ∖ { 𝑦 } ) ) |
70 |
27 68 69
|
syl2anc |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) ) → 𝐵 ≼ ( 𝐴 ∖ { 𝑦 } ) ) |
71 |
70
|
anassrs |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) → 𝐵 ≼ ( 𝐴 ∖ { 𝑦 } ) ) |
72 |
71
|
expr |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → ( 𝑥 ≠ 𝑦 → 𝐵 ≼ ( 𝐴 ∖ { 𝑦 } ) ) ) |
73 |
72
|
necon1bd |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → ( ¬ 𝐵 ≼ ( 𝐴 ∖ { 𝑦 } ) → 𝑥 = 𝑦 ) ) |
74 |
23 73
|
mpd |
⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) → 𝑥 = 𝑦 ) |
75 |
74
|
ex |
⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
76 |
75
|
ralrimivva |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
77 |
|
dff13 |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
78 |
3 76 77
|
sylanbrc |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) → 𝐹 : 𝐴 –1-1→ 𝐵 ) |
79 |
|
df-f1o |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) ) |
80 |
78 1 79
|
sylanbrc |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin ) → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |