Step |
Hyp |
Ref |
Expression |
1 |
|
aciunf1.0 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
aciunf1.1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐵 ∈ 𝑊 ) |
3 |
|
ssrab2 |
⊢ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ⊆ 𝐴 |
4 |
|
ssexg |
⊢ ( ( { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉 ) → { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ∈ V ) |
5 |
3 1 4
|
sylancr |
⊢ ( 𝜑 → { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ∈ V ) |
6 |
|
rabid |
⊢ ( 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ↔ ( 𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ) ) |
7 |
6
|
biimpi |
⊢ ( 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } → ( 𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ) ) |
8 |
7
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ) → ( 𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ) ) |
9 |
8
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ) → 𝐵 ≠ ∅ ) |
10 |
|
nfrab1 |
⊢ Ⅎ 𝑗 { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } |
11 |
8
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ) → 𝑗 ∈ 𝐴 ) |
12 |
11 2
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ) → 𝐵 ∈ 𝑊 ) |
13 |
5 9 10 12
|
aciunf1lem |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 –1-1→ ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ( { 𝑗 } × 𝐵 ) ∧ ∀ 𝑘 ∈ ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) = 𝑘 ) ) |
14 |
|
eqidd |
⊢ ( 𝜑 → 𝑓 = 𝑓 ) |
15 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
16 |
|
nfcv |
⊢ Ⅎ 𝑗 𝐴 |
17 |
|
nfrab1 |
⊢ Ⅎ 𝑗 { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } |
18 |
16 17
|
nfdif |
⊢ Ⅎ 𝑗 ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) |
19 |
|
difrab |
⊢ ( { 𝑗 ∈ 𝐴 ∣ ⊤ } ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) = { 𝑗 ∈ 𝐴 ∣ ( ⊤ ∧ ¬ 𝐵 = ∅ ) } |
20 |
16
|
rabtru |
⊢ { 𝑗 ∈ 𝐴 ∣ ⊤ } = 𝐴 |
21 |
20
|
difeq1i |
⊢ ( { 𝑗 ∈ 𝐴 ∣ ⊤ } ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) = ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) |
22 |
|
truan |
⊢ ( ( ⊤ ∧ ¬ 𝐵 = ∅ ) ↔ ¬ 𝐵 = ∅ ) |
23 |
|
df-ne |
⊢ ( 𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅ ) |
24 |
22 23
|
bitr4i |
⊢ ( ( ⊤ ∧ ¬ 𝐵 = ∅ ) ↔ 𝐵 ≠ ∅ ) |
25 |
24
|
rabbii |
⊢ { 𝑗 ∈ 𝐴 ∣ ( ⊤ ∧ ¬ 𝐵 = ∅ ) } = { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } |
26 |
19 21 25
|
3eqtr3i |
⊢ ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) = { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } |
27 |
26
|
a1i |
⊢ ( 𝜑 → ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) = { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ) |
28 |
|
eqidd |
⊢ ( 𝜑 → 𝐵 = 𝐵 ) |
29 |
15 18 10 27 28
|
iuneq12df |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) 𝐵 = ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 ) |
30 |
|
rabid |
⊢ ( 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ↔ ( 𝑗 ∈ 𝐴 ∧ 𝐵 = ∅ ) ) |
31 |
30
|
biimpi |
⊢ ( 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } → ( 𝑗 ∈ 𝐴 ∧ 𝐵 = ∅ ) ) |
32 |
31
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → ( 𝑗 ∈ 𝐴 ∧ 𝐵 = ∅ ) ) |
33 |
32
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → 𝐵 = ∅ ) |
34 |
33
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝐵 = ∅ ) |
35 |
17
|
iunxdif3 |
⊢ ( ∀ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝐵 = ∅ → ∪ 𝑗 ∈ ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) 𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵 ) |
36 |
34 35
|
syl |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) 𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵 ) |
37 |
29 36
|
eqtr3d |
⊢ ( 𝜑 → ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵 ) |
38 |
|
eqidd |
⊢ ( 𝜑 → ( { 𝑗 } × 𝐵 ) = ( { 𝑗 } × 𝐵 ) ) |
39 |
15 18 10 27 38
|
iuneq12df |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ( { 𝑗 } × 𝐵 ) = ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ( { 𝑗 } × 𝐵 ) ) |
40 |
33
|
xpeq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → ( { 𝑗 } × 𝐵 ) = ( { 𝑗 } × ∅ ) ) |
41 |
|
xp0 |
⊢ ( { 𝑗 } × ∅ ) = ∅ |
42 |
40 41
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → ( { 𝑗 } × 𝐵 ) = ∅ ) |
43 |
42
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ( { 𝑗 } × 𝐵 ) = ∅ ) |
44 |
17
|
iunxdif3 |
⊢ ( ∀ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ( { 𝑗 } × 𝐵 ) = ∅ → ∪ 𝑗 ∈ ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ( { 𝑗 } × 𝐵 ) = ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
45 |
43 44
|
syl |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ( 𝐴 ∖ { 𝑗 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ( { 𝑗 } × 𝐵 ) = ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
46 |
39 45
|
eqtr3d |
⊢ ( 𝜑 → ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ( { 𝑗 } × 𝐵 ) = ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
47 |
14 37 46
|
f1eq123d |
⊢ ( 𝜑 → ( 𝑓 : ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 –1-1→ ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ( { 𝑗 } × 𝐵 ) ↔ 𝑓 : ∪ 𝑗 ∈ 𝐴 𝐵 –1-1→ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ) |
48 |
37
|
raleqdv |
⊢ ( 𝜑 → ( ∀ 𝑘 ∈ ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) = 𝑘 ↔ ∀ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) = 𝑘 ) ) |
49 |
47 48
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝑓 : ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 –1-1→ ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ( { 𝑗 } × 𝐵 ) ∧ ∀ 𝑘 ∈ ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) = 𝑘 ) ↔ ( 𝑓 : ∪ 𝑗 ∈ 𝐴 𝐵 –1-1→ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∧ ∀ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) = 𝑘 ) ) ) |
50 |
49
|
exbidv |
⊢ ( 𝜑 → ( ∃ 𝑓 ( 𝑓 : ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 –1-1→ ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ( { 𝑗 } × 𝐵 ) ∧ ∀ 𝑘 ∈ ∪ 𝑗 ∈ { 𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } 𝐵 ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) = 𝑘 ) ↔ ∃ 𝑓 ( 𝑓 : ∪ 𝑗 ∈ 𝐴 𝐵 –1-1→ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∧ ∀ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) = 𝑘 ) ) ) |
51 |
13 50
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : ∪ 𝑗 ∈ 𝐴 𝐵 –1-1→ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∧ ∀ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) = 𝑘 ) ) |