| Step |
Hyp |
Ref |
Expression |
| 1 |
|
f1ounsn.f |
⊢ 𝐹 = ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) |
| 2 |
|
f1of |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → 𝐺 : 𝐴 ⟶ 𝐵 ) |
| 3 |
|
ssun1 |
⊢ 𝐵 ⊆ ( 𝐵 ∪ { 𝑌 } ) |
| 4 |
3
|
a1i |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → 𝐵 ⊆ ( 𝐵 ∪ { 𝑌 } ) ) |
| 5 |
2 4
|
fssd |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → 𝐺 : 𝐴 ⟶ ( 𝐵 ∪ { 𝑌 } ) ) |
| 6 |
5
|
3ad2ant1 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → 𝐺 : 𝐴 ⟶ ( 𝐵 ∪ { 𝑌 } ) ) |
| 7 |
|
simpl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → 𝑋 ∈ 𝑉 ) |
| 8 |
|
df-nel |
⊢ ( 𝑋 ∉ 𝐴 ↔ ¬ 𝑋 ∈ 𝐴 ) |
| 9 |
8
|
biimpi |
⊢ ( 𝑋 ∉ 𝐴 → ¬ 𝑋 ∈ 𝐴 ) |
| 10 |
9
|
adantr |
⊢ ( ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) → ¬ 𝑋 ∈ 𝐴 ) |
| 11 |
7 10
|
anim12i |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( 𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝐴 ) ) |
| 12 |
11
|
3adant1 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( 𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝐴 ) ) |
| 13 |
|
eqid |
⊢ 𝑌 = 𝑌 |
| 14 |
13
|
olci |
⊢ ( 𝑌 ∈ 𝐵 ∨ 𝑌 = 𝑌 ) |
| 15 |
|
elunsn |
⊢ ( 𝑌 ∈ 𝑊 → ( 𝑌 ∈ ( 𝐵 ∪ { 𝑌 } ) ↔ ( 𝑌 ∈ 𝐵 ∨ 𝑌 = 𝑌 ) ) ) |
| 16 |
14 15
|
mpbiri |
⊢ ( 𝑌 ∈ 𝑊 → 𝑌 ∈ ( 𝐵 ∪ { 𝑌 } ) ) |
| 17 |
16
|
adantl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → 𝑌 ∈ ( 𝐵 ∪ { 𝑌 } ) ) |
| 18 |
17
|
3ad2ant2 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → 𝑌 ∈ ( 𝐵 ∪ { 𝑌 } ) ) |
| 19 |
6 12 18
|
3jca |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( 𝐺 : 𝐴 ⟶ ( 𝐵 ∪ { 𝑌 } ) ∧ ( 𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝐴 ) ∧ 𝑌 ∈ ( 𝐵 ∪ { 𝑌 } ) ) ) |
| 20 |
|
fsnunf |
⊢ ( ( 𝐺 : 𝐴 ⟶ ( 𝐵 ∪ { 𝑌 } ) ∧ ( 𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝐴 ) ∧ 𝑌 ∈ ( 𝐵 ∪ { 𝑌 } ) ) → ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) ⟶ ( 𝐵 ∪ { 𝑌 } ) ) |
| 21 |
19 20
|
syl |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) ⟶ ( 𝐵 ∪ { 𝑌 } ) ) |
| 22 |
|
f1of1 |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → 𝐺 : 𝐴 –1-1→ 𝐵 ) |
| 23 |
|
dff14a |
⊢ ( 𝐺 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ≠ 𝑏 → ( 𝐺 ‘ 𝑎 ) ≠ ( 𝐺 ‘ 𝑏 ) ) ) ) |
| 24 |
|
neeq1 |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 ≠ 𝑏 ↔ 𝑥 ≠ 𝑏 ) ) |
| 25 |
|
fveq2 |
⊢ ( 𝑎 = 𝑥 → ( 𝐺 ‘ 𝑎 ) = ( 𝐺 ‘ 𝑥 ) ) |
| 26 |
25
|
neeq1d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝐺 ‘ 𝑎 ) ≠ ( 𝐺 ‘ 𝑏 ) ↔ ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑏 ) ) ) |
| 27 |
24 26
|
imbi12d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 ≠ 𝑏 → ( 𝐺 ‘ 𝑎 ) ≠ ( 𝐺 ‘ 𝑏 ) ) ↔ ( 𝑥 ≠ 𝑏 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑏 ) ) ) ) |
| 28 |
|
neeq2 |
⊢ ( 𝑏 = 𝑦 → ( 𝑥 ≠ 𝑏 ↔ 𝑥 ≠ 𝑦 ) ) |
| 29 |
|
fveq2 |
⊢ ( 𝑏 = 𝑦 → ( 𝐺 ‘ 𝑏 ) = ( 𝐺 ‘ 𝑦 ) ) |
| 30 |
29
|
neeq2d |
⊢ ( 𝑏 = 𝑦 → ( ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑏 ) ↔ ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) ) |
| 31 |
28 30
|
imbi12d |
⊢ ( 𝑏 = 𝑦 → ( ( 𝑥 ≠ 𝑏 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑏 ) ) ↔ ( 𝑥 ≠ 𝑦 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) ) ) |
| 32 |
27 31
|
rspc2va |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ≠ 𝑏 → ( 𝐺 ‘ 𝑎 ) ≠ ( 𝐺 ‘ 𝑏 ) ) ) → ( 𝑥 ≠ 𝑦 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) ) |
| 33 |
32
|
expcom |
⊢ ( ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ≠ 𝑏 → ( 𝐺 ‘ 𝑎 ) ≠ ( 𝐺 ‘ 𝑏 ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑦 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) ) ) |
| 34 |
33
|
adantl |
⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ≠ 𝑏 → ( 𝐺 ‘ 𝑎 ) ≠ ( 𝐺 ‘ 𝑏 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑦 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) ) ) |
| 35 |
23 34
|
sylbi |
⊢ ( 𝐺 : 𝐴 –1-1→ 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑦 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) ) ) |
| 36 |
22 35
|
syl |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑦 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) ) ) |
| 37 |
36
|
3ad2ant1 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑦 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) ) ) |
| 38 |
37
|
impl |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑦 → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) ) |
| 39 |
38
|
imp |
⊢ ( ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) → ( 𝐺 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑦 ) ) |
| 40 |
|
nelne2 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐴 ) → 𝑥 ≠ 𝑋 ) |
| 41 |
40
|
necomd |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐴 ) → 𝑋 ≠ 𝑥 ) |
| 42 |
41
|
expcom |
⊢ ( ¬ 𝑋 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → 𝑋 ≠ 𝑥 ) ) |
| 43 |
8 42
|
sylbi |
⊢ ( 𝑋 ∉ 𝐴 → ( 𝑥 ∈ 𝐴 → 𝑋 ≠ 𝑥 ) ) |
| 44 |
43
|
adantr |
⊢ ( ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) → ( 𝑥 ∈ 𝐴 → 𝑋 ≠ 𝑥 ) ) |
| 45 |
44
|
3ad2ant3 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( 𝑥 ∈ 𝐴 → 𝑋 ≠ 𝑥 ) ) |
| 46 |
45
|
imp |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑋 ≠ 𝑥 ) |
| 47 |
46
|
adantr |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑋 ≠ 𝑥 ) |
| 48 |
47
|
adantr |
⊢ ( ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) → 𝑋 ≠ 𝑥 ) |
| 49 |
|
fvunsn |
⊢ ( 𝑋 ≠ 𝑥 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
| 50 |
48 49
|
syl |
⊢ ( ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
| 51 |
|
nelne2 |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐴 ) → 𝑦 ≠ 𝑋 ) |
| 52 |
51
|
necomd |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐴 ) → 𝑋 ≠ 𝑦 ) |
| 53 |
52
|
expcom |
⊢ ( ¬ 𝑋 ∈ 𝐴 → ( 𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦 ) ) |
| 54 |
8 53
|
sylbi |
⊢ ( 𝑋 ∉ 𝐴 → ( 𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦 ) ) |
| 55 |
54
|
adantr |
⊢ ( ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) → ( 𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦 ) ) |
| 56 |
55
|
3ad2ant3 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( 𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦 ) ) |
| 57 |
56
|
adantr |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦 ) ) |
| 58 |
57
|
imp |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑋 ≠ 𝑦 ) |
| 59 |
58
|
adantr |
⊢ ( ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) → 𝑋 ≠ 𝑦 ) |
| 60 |
|
fvunsn |
⊢ ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
| 61 |
59 60
|
syl |
⊢ ( ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
| 62 |
39 50 61
|
3netr4d |
⊢ ( ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) |
| 63 |
62
|
ex |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 64 |
63
|
ralrimiva |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 65 |
2
|
3ad2ant1 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → 𝐺 : 𝐴 ⟶ 𝐵 ) |
| 66 |
65
|
ffvelcdmda |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐵 ) |
| 67 |
|
df-nel |
⊢ ( 𝑌 ∉ 𝐵 ↔ ¬ 𝑌 ∈ 𝐵 ) |
| 68 |
67
|
biimpi |
⊢ ( 𝑌 ∉ 𝐵 → ¬ 𝑌 ∈ 𝐵 ) |
| 69 |
68
|
adantl |
⊢ ( ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) → ¬ 𝑌 ∈ 𝐵 ) |
| 70 |
69
|
3ad2ant3 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ¬ 𝑌 ∈ 𝐵 ) |
| 71 |
70
|
adantr |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ¬ 𝑌 ∈ 𝐵 ) |
| 72 |
|
nelne2 |
⊢ ( ( ( 𝐺 ‘ 𝑥 ) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵 ) → ( 𝐺 ‘ 𝑥 ) ≠ 𝑌 ) |
| 73 |
66 71 72
|
syl2anc |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ≠ 𝑌 ) |
| 74 |
73
|
adantr |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑋 ) → ( 𝐺 ‘ 𝑥 ) ≠ 𝑌 ) |
| 75 |
|
simpr |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑋 ) → 𝑥 ≠ 𝑋 ) |
| 76 |
75
|
necomd |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑋 ) → 𝑋 ≠ 𝑥 ) |
| 77 |
76 49
|
syl |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑋 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
| 78 |
7
|
3ad2ant2 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → 𝑋 ∈ 𝑉 ) |
| 79 |
|
simpr |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → 𝑌 ∈ 𝑊 ) |
| 80 |
79
|
3ad2ant2 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → 𝑌 ∈ 𝑊 ) |
| 81 |
|
f1odm |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → dom 𝐺 = 𝐴 ) |
| 82 |
81
|
eqcomd |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → 𝐴 = dom 𝐺 ) |
| 83 |
82
|
eleq2d |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → ( 𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺 ) ) |
| 84 |
83
|
notbid |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → ( ¬ 𝑋 ∈ 𝐴 ↔ ¬ 𝑋 ∈ dom 𝐺 ) ) |
| 85 |
8 84
|
bitrid |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → ( 𝑋 ∉ 𝐴 ↔ ¬ 𝑋 ∈ dom 𝐺 ) ) |
| 86 |
85
|
biimpd |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → ( 𝑋 ∉ 𝐴 → ¬ 𝑋 ∈ dom 𝐺 ) ) |
| 87 |
86
|
adantrd |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → ( ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) → ¬ 𝑋 ∈ dom 𝐺 ) ) |
| 88 |
87
|
imp |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ¬ 𝑋 ∈ dom 𝐺 ) |
| 89 |
88
|
3adant2 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ¬ 𝑋 ∈ dom 𝐺 ) |
| 90 |
78 80 89
|
3jca |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺 ) ) |
| 91 |
90
|
adantr |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺 ) ) |
| 92 |
91
|
adantr |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑋 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺 ) ) |
| 93 |
|
fsnunfv |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) = 𝑌 ) |
| 94 |
92 93
|
syl |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑋 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) = 𝑌 ) |
| 95 |
74 77 94
|
3netr4d |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑋 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) |
| 96 |
95
|
ex |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) |
| 97 |
78
|
adantr |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑋 ∈ 𝑉 ) |
| 98 |
|
neeq2 |
⊢ ( 𝑦 = 𝑋 → ( 𝑥 ≠ 𝑦 ↔ 𝑥 ≠ 𝑋 ) ) |
| 99 |
|
fveq2 |
⊢ ( 𝑦 = 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) = ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) |
| 100 |
99
|
neeq2d |
⊢ ( 𝑦 = 𝑋 → ( ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ↔ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) |
| 101 |
98 100
|
imbi12d |
⊢ ( 𝑦 = 𝑋 → ( ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝑥 ≠ 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) ) |
| 102 |
101
|
ralsng |
⊢ ( 𝑋 ∈ 𝑉 → ( ∀ 𝑦 ∈ { 𝑋 } ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝑥 ≠ 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) ) |
| 103 |
97 102
|
syl |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ { 𝑋 } ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝑥 ≠ 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) ) |
| 104 |
96 103
|
mpbird |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑦 ∈ { 𝑋 } ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 105 |
|
ralun |
⊢ ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ { 𝑋 } ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 106 |
64 104 105
|
syl2anc |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 107 |
106
|
ralrimiva |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 108 |
65
|
ffvelcdmda |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐵 ) |
| 109 |
70
|
adantr |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) → ¬ 𝑌 ∈ 𝐵 ) |
| 110 |
108 109
|
jca |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵 ) ) |
| 111 |
110
|
adantr |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑋 ≠ 𝑦 ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵 ) ) |
| 112 |
|
nelne2 |
⊢ ( ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵 ) → ( 𝐺 ‘ 𝑦 ) ≠ 𝑌 ) |
| 113 |
112
|
necomd |
⊢ ( ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵 ) → 𝑌 ≠ ( 𝐺 ‘ 𝑦 ) ) |
| 114 |
111 113
|
syl |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑋 ≠ 𝑦 ) → 𝑌 ≠ ( 𝐺 ‘ 𝑦 ) ) |
| 115 |
90
|
adantr |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺 ) ) |
| 116 |
115
|
adantr |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑋 ≠ 𝑦 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺 ) ) |
| 117 |
116 93
|
syl |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑋 ≠ 𝑦 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) = 𝑌 ) |
| 118 |
60
|
adantl |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑋 ≠ 𝑦 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
| 119 |
114 117 118
|
3netr4d |
⊢ ( ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑋 ≠ 𝑦 ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) |
| 120 |
119
|
ex |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 121 |
120
|
ralrimiva |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ∀ 𝑦 ∈ 𝐴 ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 122 |
|
eqid |
⊢ 𝑋 = 𝑋 |
| 123 |
|
eqneqall |
⊢ ( 𝑋 = 𝑋 → ( 𝑋 ≠ 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) |
| 124 |
122 123
|
ax-mp |
⊢ ( 𝑋 ≠ 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) |
| 125 |
|
neeq2 |
⊢ ( 𝑦 = 𝑋 → ( 𝑋 ≠ 𝑦 ↔ 𝑋 ≠ 𝑋 ) ) |
| 126 |
99
|
neeq2d |
⊢ ( 𝑦 = 𝑋 → ( ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ↔ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) |
| 127 |
125 126
|
imbi12d |
⊢ ( 𝑦 = 𝑋 → ( ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝑋 ≠ 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) ) |
| 128 |
127
|
ralsng |
⊢ ( 𝑋 ∈ 𝑉 → ( ∀ 𝑦 ∈ { 𝑋 } ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝑋 ≠ 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) ) |
| 129 |
78 128
|
syl |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( ∀ 𝑦 ∈ { 𝑋 } ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝑋 ≠ 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) ) ) |
| 130 |
124 129
|
mpbiri |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ∀ 𝑦 ∈ { 𝑋 } ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 131 |
|
ralun |
⊢ ( ( ∀ 𝑦 ∈ 𝐴 ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ { 𝑋 } ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 132 |
121 130 131
|
syl2anc |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 133 |
|
neeq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≠ 𝑦 ↔ 𝑋 ≠ 𝑦 ) ) |
| 134 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) = ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ) |
| 135 |
134
|
neeq1d |
⊢ ( 𝑥 = 𝑋 → ( ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ↔ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 136 |
133 135
|
imbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) ) |
| 137 |
136
|
ralbidv |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) ) |
| 138 |
137
|
ralsng |
⊢ ( 𝑋 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝑋 } ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) ) |
| 139 |
78 138
|
syl |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( ∀ 𝑥 ∈ { 𝑋 } ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑋 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑋 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) ) |
| 140 |
132 139
|
mpbird |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ∀ 𝑥 ∈ { 𝑋 } ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 141 |
|
ralun |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ∧ ∀ 𝑥 ∈ { 𝑋 } ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) → ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝑋 } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 142 |
107 140 141
|
syl2anc |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝑋 } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) |
| 143 |
21 142
|
jca |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) ⟶ ( 𝐵 ∪ { 𝑌 } ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝑋 } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) ) |
| 144 |
|
rnun |
⊢ ran ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) = ( ran 𝐺 ∪ ran { 〈 𝑋 , 𝑌 〉 } ) |
| 145 |
|
f1ofo |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → 𝐺 : 𝐴 –onto→ 𝐵 ) |
| 146 |
|
forn |
⊢ ( 𝐺 : 𝐴 –onto→ 𝐵 → ran 𝐺 = 𝐵 ) |
| 147 |
145 146
|
syl |
⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 → ran 𝐺 = 𝐵 ) |
| 148 |
147
|
3ad2ant1 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ran 𝐺 = 𝐵 ) |
| 149 |
|
rnsnopg |
⊢ ( 𝑋 ∈ 𝑉 → ran { 〈 𝑋 , 𝑌 〉 } = { 𝑌 } ) |
| 150 |
149
|
adantr |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → ran { 〈 𝑋 , 𝑌 〉 } = { 𝑌 } ) |
| 151 |
150
|
3ad2ant2 |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ran { 〈 𝑋 , 𝑌 〉 } = { 𝑌 } ) |
| 152 |
148 151
|
uneq12d |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( ran 𝐺 ∪ ran { 〈 𝑋 , 𝑌 〉 } ) = ( 𝐵 ∪ { 𝑌 } ) ) |
| 153 |
144 152
|
eqtrid |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ran ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) = ( 𝐵 ∪ { 𝑌 } ) ) |
| 154 |
143 153
|
jca |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) ⟶ ( 𝐵 ∪ { 𝑌 } ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝑋 } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) ∧ ran ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) = ( 𝐵 ∪ { 𝑌 } ) ) ) |
| 155 |
|
dff1o5 |
⊢ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) –1-1-onto→ ( 𝐵 ∪ { 𝑌 } ) ↔ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) –1-1→ ( 𝐵 ∪ { 𝑌 } ) ∧ ran ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) = ( 𝐵 ∪ { 𝑌 } ) ) ) |
| 156 |
|
dff14a |
⊢ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) –1-1→ ( 𝐵 ∪ { 𝑌 } ) ↔ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) ⟶ ( 𝐵 ∪ { 𝑌 } ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝑋 } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) ) |
| 157 |
155 156
|
bianbi |
⊢ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) –1-1-onto→ ( 𝐵 ∪ { 𝑌 } ) ↔ ( ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) ⟶ ( 𝐵 ∪ { 𝑌 } ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝑋 } ) ∀ 𝑦 ∈ ( 𝐴 ∪ { 𝑋 } ) ( 𝑥 ≠ 𝑦 → ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑥 ) ≠ ( ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) ‘ 𝑦 ) ) ) ∧ ran ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) = ( 𝐵 ∪ { 𝑌 } ) ) ) |
| 158 |
154 157
|
sylibr |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) –1-1-onto→ ( 𝐵 ∪ { 𝑌 } ) ) |
| 159 |
|
f1oeq1 |
⊢ ( 𝐹 = ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) → ( 𝐹 : ( 𝐴 ∪ { 𝑋 } ) –1-1-onto→ ( 𝐵 ∪ { 𝑌 } ) ↔ ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) –1-1-onto→ ( 𝐵 ∪ { 𝑌 } ) ) ) |
| 160 |
1 159
|
ax-mp |
⊢ ( 𝐹 : ( 𝐴 ∪ { 𝑋 } ) –1-1-onto→ ( 𝐵 ∪ { 𝑌 } ) ↔ ( 𝐺 ∪ { 〈 𝑋 , 𝑌 〉 } ) : ( 𝐴 ∪ { 𝑋 } ) –1-1-onto→ ( 𝐵 ∪ { 𝑌 } ) ) |
| 161 |
158 160
|
sylibr |
⊢ ( ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵 ) ) → 𝐹 : ( 𝐴 ∪ { 𝑋 } ) –1-1-onto→ ( 𝐵 ∪ { 𝑌 } ) ) |