Step |
Hyp |
Ref |
Expression |
1 |
|
mapfzcons.1 |
⊢ 𝑀 = ( 𝑁 + 1 ) |
2 |
|
ovex |
⊢ ( 𝑁 + 1 ) ∈ V |
3 |
1 2
|
eqeltri |
⊢ 𝑀 ∈ V |
4 |
3
|
a1i |
⊢ ( ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → 𝑀 ∈ V ) |
5 |
|
elex |
⊢ ( 𝐶 ∈ 𝐵 → 𝐶 ∈ V ) |
6 |
5
|
adantl |
⊢ ( ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → 𝐶 ∈ V ) |
7 |
|
elmapi |
⊢ ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) → 𝐴 : ( 1 ... 𝑁 ) ⟶ 𝐵 ) |
8 |
7
|
fdmd |
⊢ ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) → dom 𝐴 = ( 1 ... 𝑁 ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → dom 𝐴 = ( 1 ... 𝑁 ) ) |
10 |
9
|
ineq1d |
⊢ ( ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( dom 𝐴 ∩ { 𝑀 } ) = ( ( 1 ... 𝑁 ) ∩ { 𝑀 } ) ) |
11 |
1
|
sneqi |
⊢ { 𝑀 } = { ( 𝑁 + 1 ) } |
12 |
11
|
ineq2i |
⊢ ( ( 1 ... 𝑁 ) ∩ { 𝑀 } ) = ( ( 1 ... 𝑁 ) ∩ { ( 𝑁 + 1 ) } ) |
13 |
|
fzp1disj |
⊢ ( ( 1 ... 𝑁 ) ∩ { ( 𝑁 + 1 ) } ) = ∅ |
14 |
12 13
|
eqtri |
⊢ ( ( 1 ... 𝑁 ) ∩ { 𝑀 } ) = ∅ |
15 |
10 14
|
eqtrdi |
⊢ ( ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( dom 𝐴 ∩ { 𝑀 } ) = ∅ ) |
16 |
|
disjsn |
⊢ ( ( dom 𝐴 ∩ { 𝑀 } ) = ∅ ↔ ¬ 𝑀 ∈ dom 𝐴 ) |
17 |
15 16
|
sylib |
⊢ ( ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ¬ 𝑀 ∈ dom 𝐴 ) |
18 |
|
fsnunfv |
⊢ ( ( 𝑀 ∈ V ∧ 𝐶 ∈ V ∧ ¬ 𝑀 ∈ dom 𝐴 ) → ( ( 𝐴 ∪ { 〈 𝑀 , 𝐶 〉 } ) ‘ 𝑀 ) = 𝐶 ) |
19 |
4 6 17 18
|
syl3anc |
⊢ ( ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( ( 𝐴 ∪ { 〈 𝑀 , 𝐶 〉 } ) ‘ 𝑀 ) = 𝐶 ) |