Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ) |
2 |
1
|
anim2i |
⊢ ( ( 𝑍 ∈ V ∧ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) ) → ( 𝑍 ∈ V ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ) ) |
3 |
2
|
ancomd |
⊢ ( ( 𝑍 ∈ V ∧ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) ) → ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ 𝑍 ∈ V ) ) |
4 |
|
df-3an |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ V ) ↔ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ 𝑍 ∈ V ) ) |
5 |
3 4
|
sylibr |
⊢ ( ( 𝑍 ∈ V ∧ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ V ) ) |
6 |
|
snopfsupp |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ V ) → { 〈 𝑋 , 𝑌 〉 } finSupp 𝑍 ) |
7 |
5 6
|
syl |
⊢ ( ( 𝑍 ∈ V ∧ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) ) → { 〈 𝑋 , 𝑌 〉 } finSupp 𝑍 ) |
8 |
|
funsng |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → Fun { 〈 𝑋 , 𝑌 〉 } ) |
9 |
|
simpl |
⊢ ( ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) → Fun 𝐹 ) |
10 |
8 9
|
anim12ci |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) → ( Fun 𝐹 ∧ Fun { 〈 𝑋 , 𝑌 〉 } ) ) |
11 |
|
dmsnopg |
⊢ ( 𝑌 ∈ 𝑊 → dom { 〈 𝑋 , 𝑌 〉 } = { 𝑋 } ) |
12 |
11
|
adantl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → dom { 〈 𝑋 , 𝑌 〉 } = { 𝑋 } ) |
13 |
12
|
ineq2d |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → ( dom 𝐹 ∩ dom { 〈 𝑋 , 𝑌 〉 } ) = ( dom 𝐹 ∩ { 𝑋 } ) ) |
14 |
|
df-nel |
⊢ ( 𝑋 ∉ dom 𝐹 ↔ ¬ 𝑋 ∈ dom 𝐹 ) |
15 |
|
disjsn |
⊢ ( ( dom 𝐹 ∩ { 𝑋 } ) = ∅ ↔ ¬ 𝑋 ∈ dom 𝐹 ) |
16 |
14 15
|
sylbb2 |
⊢ ( 𝑋 ∉ dom 𝐹 → ( dom 𝐹 ∩ { 𝑋 } ) = ∅ ) |
17 |
16
|
adantl |
⊢ ( ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) → ( dom 𝐹 ∩ { 𝑋 } ) = ∅ ) |
18 |
13 17
|
sylan9eq |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) → ( dom 𝐹 ∩ dom { 〈 𝑋 , 𝑌 〉 } ) = ∅ ) |
19 |
10 18
|
jca |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) → ( ( Fun 𝐹 ∧ Fun { 〈 𝑋 , 𝑌 〉 } ) ∧ ( dom 𝐹 ∩ dom { 〈 𝑋 , 𝑌 〉 } ) = ∅ ) ) |
20 |
19
|
adantl |
⊢ ( ( 𝑍 ∈ V ∧ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) ) → ( ( Fun 𝐹 ∧ Fun { 〈 𝑋 , 𝑌 〉 } ) ∧ ( dom 𝐹 ∩ dom { 〈 𝑋 , 𝑌 〉 } ) = ∅ ) ) |
21 |
|
funun |
⊢ ( ( ( Fun 𝐹 ∧ Fun { 〈 𝑋 , 𝑌 〉 } ) ∧ ( dom 𝐹 ∩ dom { 〈 𝑋 , 𝑌 〉 } ) = ∅ ) → Fun ( 𝐹 ∪ { 〈 𝑋 , 𝑌 〉 } ) ) |
22 |
20 21
|
syl |
⊢ ( ( 𝑍 ∈ V ∧ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) ) → Fun ( 𝐹 ∪ { 〈 𝑋 , 𝑌 〉 } ) ) |
23 |
22
|
fsuppunbi |
⊢ ( ( 𝑍 ∈ V ∧ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) ) → ( ( 𝐹 ∪ { 〈 𝑋 , 𝑌 〉 } ) finSupp 𝑍 ↔ ( 𝐹 finSupp 𝑍 ∧ { 〈 𝑋 , 𝑌 〉 } finSupp 𝑍 ) ) ) |
24 |
7 23
|
mpbiran2d |
⊢ ( ( 𝑍 ∈ V ∧ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) ) → ( ( 𝐹 ∪ { 〈 𝑋 , 𝑌 〉 } ) finSupp 𝑍 ↔ 𝐹 finSupp 𝑍 ) ) |
25 |
24
|
ex |
⊢ ( 𝑍 ∈ V → ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) → ( ( 𝐹 ∪ { 〈 𝑋 , 𝑌 〉 } ) finSupp 𝑍 ↔ 𝐹 finSupp 𝑍 ) ) ) |
26 |
|
relfsupp |
⊢ Rel finSupp |
27 |
26
|
brrelex2i |
⊢ ( ( 𝐹 ∪ { 〈 𝑋 , 𝑌 〉 } ) finSupp 𝑍 → 𝑍 ∈ V ) |
28 |
26
|
brrelex2i |
⊢ ( 𝐹 finSupp 𝑍 → 𝑍 ∈ V ) |
29 |
27 28
|
pm5.21ni |
⊢ ( ¬ 𝑍 ∈ V → ( ( 𝐹 ∪ { 〈 𝑋 , 𝑌 〉 } ) finSupp 𝑍 ↔ 𝐹 finSupp 𝑍 ) ) |
30 |
29
|
a1d |
⊢ ( ¬ 𝑍 ∈ V → ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) → ( ( 𝐹 ∪ { 〈 𝑋 , 𝑌 〉 } ) finSupp 𝑍 ↔ 𝐹 finSupp 𝑍 ) ) ) |
31 |
25 30
|
pm2.61i |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹 ) ) → ( ( 𝐹 ∪ { 〈 𝑋 , 𝑌 〉 } ) finSupp 𝑍 ↔ 𝐹 finSupp 𝑍 ) ) |