Step |
Hyp |
Ref |
Expression |
1 |
|
metakunt17.1 |
⊢ ( 𝜑 → 𝐺 : 𝐴 –1-1-onto→ 𝑋 ) |
2 |
|
metakunt17.2 |
⊢ ( 𝜑 → 𝐻 : 𝐵 –1-1-onto→ 𝑌 ) |
3 |
|
metakunt17.3 |
⊢ ( 𝜑 → 𝐼 : 𝐶 –1-1-onto→ 𝑍 ) |
4 |
|
metakunt17.4 |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
5 |
|
metakunt17.5 |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) = ∅ ) |
6 |
|
metakunt17.6 |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐶 ) = ∅ ) |
7 |
|
metakunt17.7 |
⊢ ( 𝜑 → ( 𝑋 ∩ 𝑌 ) = ∅ ) |
8 |
|
metakunt17.8 |
⊢ ( 𝜑 → ( 𝑋 ∩ 𝑍 ) = ∅ ) |
9 |
|
metakunt17.9 |
⊢ ( 𝜑 → ( 𝑌 ∩ 𝑍 ) = ∅ ) |
10 |
|
metakunt17.10 |
⊢ ( 𝜑 → 𝐹 = ( ( 𝐺 ∪ 𝐻 ) ∪ 𝐼 ) ) |
11 |
|
metakunt17.11 |
⊢ ( 𝜑 → 𝐷 = ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
12 |
|
metakunt17.12 |
⊢ ( 𝜑 → 𝑊 = ( ( 𝑋 ∪ 𝑌 ) ∪ 𝑍 ) ) |
13 |
4 7
|
jca |
⊢ ( 𝜑 → ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ ( 𝑋 ∩ 𝑌 ) = ∅ ) ) |
14 |
1 2 13
|
jca31 |
⊢ ( 𝜑 → ( ( 𝐺 : 𝐴 –1-1-onto→ 𝑋 ∧ 𝐻 : 𝐵 –1-1-onto→ 𝑌 ) ∧ ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ ( 𝑋 ∩ 𝑌 ) = ∅ ) ) ) |
15 |
|
f1oun |
⊢ ( ( ( 𝐺 : 𝐴 –1-1-onto→ 𝑋 ∧ 𝐻 : 𝐵 –1-1-onto→ 𝑌 ) ∧ ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ ( 𝑋 ∩ 𝑌 ) = ∅ ) ) → ( 𝐺 ∪ 𝐻 ) : ( 𝐴 ∪ 𝐵 ) –1-1-onto→ ( 𝑋 ∪ 𝑌 ) ) |
16 |
14 15
|
syl |
⊢ ( 𝜑 → ( 𝐺 ∪ 𝐻 ) : ( 𝐴 ∪ 𝐵 ) –1-1-onto→ ( 𝑋 ∪ 𝑌 ) ) |
17 |
|
indir |
⊢ ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ 𝐶 ) ) |
18 |
5 6
|
uneq12d |
⊢ ( 𝜑 → ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ 𝐶 ) ) = ( ∅ ∪ ∅ ) ) |
19 |
|
0un |
⊢ ( ∅ ∪ ∅ ) = ∅ |
20 |
19
|
a1i |
⊢ ( 𝜑 → ( ∅ ∪ ∅ ) = ∅ ) |
21 |
18 20
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ 𝐶 ) ) = ∅ ) |
22 |
17 21
|
syl5eq |
⊢ ( 𝜑 → ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ∅ ) |
23 |
|
indir |
⊢ ( ( 𝑋 ∪ 𝑌 ) ∩ 𝑍 ) = ( ( 𝑋 ∩ 𝑍 ) ∪ ( 𝑌 ∩ 𝑍 ) ) |
24 |
8 9
|
uneq12d |
⊢ ( 𝜑 → ( ( 𝑋 ∩ 𝑍 ) ∪ ( 𝑌 ∩ 𝑍 ) ) = ( ∅ ∪ ∅ ) ) |
25 |
24 20
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑋 ∩ 𝑍 ) ∪ ( 𝑌 ∩ 𝑍 ) ) = ∅ ) |
26 |
23 25
|
syl5eq |
⊢ ( 𝜑 → ( ( 𝑋 ∪ 𝑌 ) ∩ 𝑍 ) = ∅ ) |
27 |
22 26
|
jca |
⊢ ( 𝜑 → ( ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ∅ ∧ ( ( 𝑋 ∪ 𝑌 ) ∩ 𝑍 ) = ∅ ) ) |
28 |
16 3 27
|
jca31 |
⊢ ( 𝜑 → ( ( ( 𝐺 ∪ 𝐻 ) : ( 𝐴 ∪ 𝐵 ) –1-1-onto→ ( 𝑋 ∪ 𝑌 ) ∧ 𝐼 : 𝐶 –1-1-onto→ 𝑍 ) ∧ ( ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ∅ ∧ ( ( 𝑋 ∪ 𝑌 ) ∩ 𝑍 ) = ∅ ) ) ) |
29 |
|
f1oun |
⊢ ( ( ( ( 𝐺 ∪ 𝐻 ) : ( 𝐴 ∪ 𝐵 ) –1-1-onto→ ( 𝑋 ∪ 𝑌 ) ∧ 𝐼 : 𝐶 –1-1-onto→ 𝑍 ) ∧ ( ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ∅ ∧ ( ( 𝑋 ∪ 𝑌 ) ∩ 𝑍 ) = ∅ ) ) → ( ( 𝐺 ∪ 𝐻 ) ∪ 𝐼 ) : ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) –1-1-onto→ ( ( 𝑋 ∪ 𝑌 ) ∪ 𝑍 ) ) |
30 |
28 29
|
syl |
⊢ ( 𝜑 → ( ( 𝐺 ∪ 𝐻 ) ∪ 𝐼 ) : ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) –1-1-onto→ ( ( 𝑋 ∪ 𝑌 ) ∪ 𝑍 ) ) |
31 |
10 11 12
|
f1oeq123d |
⊢ ( 𝜑 → ( 𝐹 : 𝐷 –1-1-onto→ 𝑊 ↔ ( ( 𝐺 ∪ 𝐻 ) ∪ 𝐼 ) : ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) –1-1-onto→ ( ( 𝑋 ∪ 𝑌 ) ∪ 𝑍 ) ) ) |
32 |
30 31
|
mpbird |
⊢ ( 𝜑 → 𝐹 : 𝐷 –1-1-onto→ 𝑊 ) |