Step |
Hyp |
Ref |
Expression |
1 |
|
mapfzcons.1 |
⊢ 𝑀 = ( 𝑁 + 1 ) |
2 |
|
simp2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ) |
3 |
|
elmapex |
⊢ ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) → ( 𝐵 ∈ V ∧ ( 1 ... 𝑁 ) ∈ V ) ) |
4 |
3
|
simpld |
⊢ ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) → 𝐵 ∈ V ) |
5 |
4
|
3ad2ant2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → 𝐵 ∈ V ) |
6 |
|
ovex |
⊢ ( 1 ... 𝑁 ) ∈ V |
7 |
|
elmapg |
⊢ ( ( 𝐵 ∈ V ∧ ( 1 ... 𝑁 ) ∈ V ) → ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ↔ 𝐴 : ( 1 ... 𝑁 ) ⟶ 𝐵 ) ) |
8 |
5 6 7
|
sylancl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ↔ 𝐴 : ( 1 ... 𝑁 ) ⟶ 𝐵 ) ) |
9 |
2 8
|
mpbid |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → 𝐴 : ( 1 ... 𝑁 ) ⟶ 𝐵 ) |
10 |
|
ovex |
⊢ ( 𝑁 + 1 ) ∈ V |
11 |
|
simp3 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → 𝐶 ∈ 𝐵 ) |
12 |
|
f1osng |
⊢ ( ( ( 𝑁 + 1 ) ∈ V ∧ 𝐶 ∈ 𝐵 ) → { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } : { ( 𝑁 + 1 ) } –1-1-onto→ { 𝐶 } ) |
13 |
10 11 12
|
sylancr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } : { ( 𝑁 + 1 ) } –1-1-onto→ { 𝐶 } ) |
14 |
|
f1of |
⊢ ( { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } : { ( 𝑁 + 1 ) } –1-1-onto→ { 𝐶 } → { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } : { ( 𝑁 + 1 ) } ⟶ { 𝐶 } ) |
15 |
13 14
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } : { ( 𝑁 + 1 ) } ⟶ { 𝐶 } ) |
16 |
|
snssi |
⊢ ( 𝐶 ∈ 𝐵 → { 𝐶 } ⊆ 𝐵 ) |
17 |
16
|
3ad2ant3 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → { 𝐶 } ⊆ 𝐵 ) |
18 |
15 17
|
fssd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } : { ( 𝑁 + 1 ) } ⟶ 𝐵 ) |
19 |
|
fzp1disj |
⊢ ( ( 1 ... 𝑁 ) ∩ { ( 𝑁 + 1 ) } ) = ∅ |
20 |
19
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( ( 1 ... 𝑁 ) ∩ { ( 𝑁 + 1 ) } ) = ∅ ) |
21 |
|
fun |
⊢ ( ( ( 𝐴 : ( 1 ... 𝑁 ) ⟶ 𝐵 ∧ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } : { ( 𝑁 + 1 ) } ⟶ 𝐵 ) ∧ ( ( 1 ... 𝑁 ) ∩ { ( 𝑁 + 1 ) } ) = ∅ ) → ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) : ( ( 1 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ⟶ ( 𝐵 ∪ 𝐵 ) ) |
22 |
9 18 20 21
|
syl21anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) : ( ( 1 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ⟶ ( 𝐵 ∪ 𝐵 ) ) |
23 |
|
1z |
⊢ 1 ∈ ℤ |
24 |
|
simp1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → 𝑁 ∈ ℕ0 ) |
25 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
26 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
27 |
26
|
fveq2i |
⊢ ( ℤ≥ ‘ ( 1 − 1 ) ) = ( ℤ≥ ‘ 0 ) |
28 |
25 27
|
eqtr4i |
⊢ ℕ0 = ( ℤ≥ ‘ ( 1 − 1 ) ) |
29 |
24 28
|
eleqtrdi |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → 𝑁 ∈ ( ℤ≥ ‘ ( 1 − 1 ) ) ) |
30 |
|
fzsuc2 |
⊢ ( ( 1 ∈ ℤ ∧ 𝑁 ∈ ( ℤ≥ ‘ ( 1 − 1 ) ) ) → ( 1 ... ( 𝑁 + 1 ) ) = ( ( 1 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ) |
31 |
23 29 30
|
sylancr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( 1 ... ( 𝑁 + 1 ) ) = ( ( 1 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ) |
32 |
31
|
eqcomd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( ( 1 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) = ( 1 ... ( 𝑁 + 1 ) ) ) |
33 |
|
unidm |
⊢ ( 𝐵 ∪ 𝐵 ) = 𝐵 |
34 |
33
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝐵 ∪ 𝐵 ) = 𝐵 ) |
35 |
32 34
|
feq23d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) : ( ( 1 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ⟶ ( 𝐵 ∪ 𝐵 ) ↔ ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ 𝐵 ) ) |
36 |
22 35
|
mpbid |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ 𝐵 ) |
37 |
|
ovex |
⊢ ( 1 ... ( 𝑁 + 1 ) ) ∈ V |
38 |
|
elmapg |
⊢ ( ( 𝐵 ∈ V ∧ ( 1 ... ( 𝑁 + 1 ) ) ∈ V ) → ( ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) ∈ ( 𝐵 ↑m ( 1 ... ( 𝑁 + 1 ) ) ) ↔ ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ 𝐵 ) ) |
39 |
5 37 38
|
sylancl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) ∈ ( 𝐵 ↑m ( 1 ... ( 𝑁 + 1 ) ) ) ↔ ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ 𝐵 ) ) |
40 |
36 39
|
mpbird |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) ∈ ( 𝐵 ↑m ( 1 ... ( 𝑁 + 1 ) ) ) ) |
41 |
1
|
opeq1i |
⊢ 〈 𝑀 , 𝐶 〉 = 〈 ( 𝑁 + 1 ) , 𝐶 〉 |
42 |
41
|
sneqi |
⊢ { 〈 𝑀 , 𝐶 〉 } = { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } |
43 |
42
|
uneq2i |
⊢ ( 𝐴 ∪ { 〈 𝑀 , 𝐶 〉 } ) = ( 𝐴 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) |
44 |
1
|
oveq2i |
⊢ ( 1 ... 𝑀 ) = ( 1 ... ( 𝑁 + 1 ) ) |
45 |
44
|
oveq2i |
⊢ ( 𝐵 ↑m ( 1 ... 𝑀 ) ) = ( 𝐵 ↑m ( 1 ... ( 𝑁 + 1 ) ) ) |
46 |
40 43 45
|
3eltr4g |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ( 𝐵 ↑m ( 1 ... 𝑁 ) ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝐴 ∪ { 〈 𝑀 , 𝐶 〉 } ) ∈ ( 𝐵 ↑m ( 1 ... 𝑀 ) ) ) |