Step |
Hyp |
Ref |
Expression |
1 |
|
rnfi |
⊢ ( 𝐹 ∈ Fin → ran 𝐹 ∈ Fin ) |
2 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) ∧ ran 𝐹 ∈ Fin ) → ran 𝐹 ∈ Fin ) |
3 |
|
f1dm |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → dom 𝐹 = 𝐴 ) |
4 |
|
f1f1orn |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 ) |
5 |
|
eleq1 |
⊢ ( 𝐴 = dom 𝐹 → ( 𝐴 ∈ 𝑉 ↔ dom 𝐹 ∈ 𝑉 ) ) |
6 |
|
f1oeq2 |
⊢ ( 𝐴 = dom 𝐹 → ( 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 ↔ 𝐹 : dom 𝐹 –1-1-onto→ ran 𝐹 ) ) |
7 |
5 6
|
anbi12d |
⊢ ( 𝐴 = dom 𝐹 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 ) ↔ ( dom 𝐹 ∈ 𝑉 ∧ 𝐹 : dom 𝐹 –1-1-onto→ ran 𝐹 ) ) ) |
8 |
7
|
eqcoms |
⊢ ( dom 𝐹 = 𝐴 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 ) ↔ ( dom 𝐹 ∈ 𝑉 ∧ 𝐹 : dom 𝐹 –1-1-onto→ ran 𝐹 ) ) ) |
9 |
8
|
biimpd |
⊢ ( dom 𝐹 = 𝐴 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 ) → ( dom 𝐹 ∈ 𝑉 ∧ 𝐹 : dom 𝐹 –1-1-onto→ ran 𝐹 ) ) ) |
10 |
9
|
expcomd |
⊢ ( dom 𝐹 = 𝐴 → ( 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 → ( 𝐴 ∈ 𝑉 → ( dom 𝐹 ∈ 𝑉 ∧ 𝐹 : dom 𝐹 –1-1-onto→ ran 𝐹 ) ) ) ) |
11 |
3 4 10
|
sylc |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( 𝐴 ∈ 𝑉 → ( dom 𝐹 ∈ 𝑉 ∧ 𝐹 : dom 𝐹 –1-1-onto→ ran 𝐹 ) ) ) |
12 |
11
|
impcom |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( dom 𝐹 ∈ 𝑉 ∧ 𝐹 : dom 𝐹 –1-1-onto→ ran 𝐹 ) ) |
13 |
12
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) ∧ ran 𝐹 ∈ Fin ) → ( dom 𝐹 ∈ 𝑉 ∧ 𝐹 : dom 𝐹 –1-1-onto→ ran 𝐹 ) ) |
14 |
|
f1oeng |
⊢ ( ( dom 𝐹 ∈ 𝑉 ∧ 𝐹 : dom 𝐹 –1-1-onto→ ran 𝐹 ) → dom 𝐹 ≈ ran 𝐹 ) |
15 |
13 14
|
syl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) ∧ ran 𝐹 ∈ Fin ) → dom 𝐹 ≈ ran 𝐹 ) |
16 |
|
enfii |
⊢ ( ( ran 𝐹 ∈ Fin ∧ dom 𝐹 ≈ ran 𝐹 ) → dom 𝐹 ∈ Fin ) |
17 |
2 15 16
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) ∧ ran 𝐹 ∈ Fin ) → dom 𝐹 ∈ Fin ) |
18 |
|
f1fun |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → Fun 𝐹 ) |
19 |
18
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) ∧ ran 𝐹 ∈ Fin ) → Fun 𝐹 ) |
20 |
|
fundmfibi |
⊢ ( Fun 𝐹 → ( 𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin ) ) |
21 |
19 20
|
syl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) ∧ ran 𝐹 ∈ Fin ) → ( 𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin ) ) |
22 |
17 21
|
mpbird |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) ∧ ran 𝐹 ∈ Fin ) → 𝐹 ∈ Fin ) |
23 |
22
|
ex |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( ran 𝐹 ∈ Fin → 𝐹 ∈ Fin ) ) |
24 |
1 23
|
impbid2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( 𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin ) ) |