Step |
Hyp |
Ref |
Expression |
1 |
|
fiun.1 |
⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 ) |
2 |
|
fiun.2 |
⊢ 𝐵 ∈ V |
3 |
|
vex |
⊢ 𝑢 ∈ V |
4 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑢 → ( 𝑧 = 𝐵 ↔ 𝑢 = 𝐵 ) ) |
5 |
4
|
rexbidv |
⊢ ( 𝑧 = 𝑢 → ( ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑢 = 𝐵 ) ) |
6 |
3 5
|
elab |
⊢ ( 𝑢 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ↔ ∃ 𝑥 ∈ 𝐴 𝑢 = 𝐵 ) |
7 |
|
r19.29 |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) ∧ ∃ 𝑥 ∈ 𝐴 𝑢 = 𝐵 ) → ∃ 𝑥 ∈ 𝐴 ( ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) ∧ 𝑢 = 𝐵 ) ) |
8 |
|
nfv |
⊢ Ⅎ 𝑥 Fun 𝑢 |
9 |
|
nfre1 |
⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 |
10 |
9
|
nfab |
⊢ Ⅎ 𝑥 { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } |
11 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) |
12 |
10 11
|
nfralw |
⊢ Ⅎ 𝑥 ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) |
13 |
8 12
|
nfan |
⊢ Ⅎ 𝑥 ( Fun 𝑢 ∧ ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) ) |
14 |
|
ffun |
⊢ ( 𝐵 : 𝐷 ⟶ 𝑆 → Fun 𝐵 ) |
15 |
|
funeq |
⊢ ( 𝑢 = 𝐵 → ( Fun 𝑢 ↔ Fun 𝐵 ) ) |
16 |
|
bianir |
⊢ ( ( Fun 𝐵 ∧ ( Fun 𝑢 ↔ Fun 𝐵 ) ) → Fun 𝑢 ) |
17 |
14 15 16
|
syl2an |
⊢ ( ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ 𝑢 = 𝐵 ) → Fun 𝑢 ) |
18 |
17
|
adantlr |
⊢ ( ( ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) ∧ 𝑢 = 𝐵 ) → Fun 𝑢 ) |
19 |
1
|
fiunlem |
⊢ ( ( ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) ∧ 𝑢 = 𝐵 ) → ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) ) |
20 |
18 19
|
jca |
⊢ ( ( ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) ∧ 𝑢 = 𝐵 ) → ( Fun 𝑢 ∧ ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) ) ) |
21 |
20
|
a1i |
⊢ ( 𝑥 ∈ 𝐴 → ( ( ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) ∧ 𝑢 = 𝐵 ) → ( Fun 𝑢 ∧ ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) ) ) ) |
22 |
13 21
|
rexlimi |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) ∧ 𝑢 = 𝐵 ) → ( Fun 𝑢 ∧ ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) ) ) |
23 |
7 22
|
syl |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) ∧ ∃ 𝑥 ∈ 𝐴 𝑢 = 𝐵 ) → ( Fun 𝑢 ∧ ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) ) ) |
24 |
6 23
|
sylan2b |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) ∧ 𝑢 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ) → ( Fun 𝑢 ∧ ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) ) ) |
25 |
24
|
ralrimiva |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ∀ 𝑢 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( Fun 𝑢 ∧ ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) ) ) |
26 |
|
fununi |
⊢ ( ∀ 𝑢 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( Fun 𝑢 ∧ ∀ 𝑣 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ( 𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢 ) ) → Fun ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ) |
27 |
25 26
|
syl |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → Fun ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ) |
28 |
2
|
dfiun2 |
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } |
29 |
28
|
funeqi |
⊢ ( Fun ∪ 𝑥 ∈ 𝐴 𝐵 ↔ Fun ∪ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ) |
30 |
27 29
|
sylibr |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → Fun ∪ 𝑥 ∈ 𝐴 𝐵 ) |
31 |
3
|
eldm2 |
⊢ ( 𝑢 ∈ dom 𝐵 ↔ ∃ 𝑣 〈 𝑢 , 𝑣 〉 ∈ 𝐵 ) |
32 |
|
fdm |
⊢ ( 𝐵 : 𝐷 ⟶ 𝑆 → dom 𝐵 = 𝐷 ) |
33 |
32
|
eleq2d |
⊢ ( 𝐵 : 𝐷 ⟶ 𝑆 → ( 𝑢 ∈ dom 𝐵 ↔ 𝑢 ∈ 𝐷 ) ) |
34 |
31 33
|
bitr3id |
⊢ ( 𝐵 : 𝐷 ⟶ 𝑆 → ( ∃ 𝑣 〈 𝑢 , 𝑣 〉 ∈ 𝐵 ↔ 𝑢 ∈ 𝐷 ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ( ∃ 𝑣 〈 𝑢 , 𝑣 〉 ∈ 𝐵 ↔ 𝑢 ∈ 𝐷 ) ) |
36 |
35
|
ralrexbid |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑣 〈 𝑢 , 𝑣 〉 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑢 ∈ 𝐷 ) ) |
37 |
|
eliun |
⊢ ( 〈 𝑢 , 𝑣 〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 〈 𝑢 , 𝑣 〉 ∈ 𝐵 ) |
38 |
37
|
exbii |
⊢ ( ∃ 𝑣 〈 𝑢 , 𝑣 〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑣 ∃ 𝑥 ∈ 𝐴 〈 𝑢 , 𝑣 〉 ∈ 𝐵 ) |
39 |
3
|
eldm2 |
⊢ ( 𝑢 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑣 〈 𝑢 , 𝑣 〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
40 |
|
rexcom4 |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑣 〈 𝑢 , 𝑣 〉 ∈ 𝐵 ↔ ∃ 𝑣 ∃ 𝑥 ∈ 𝐴 〈 𝑢 , 𝑣 〉 ∈ 𝐵 ) |
41 |
38 39 40
|
3bitr4i |
⊢ ( 𝑢 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑣 〈 𝑢 , 𝑣 〉 ∈ 𝐵 ) |
42 |
|
eliun |
⊢ ( 𝑢 ∈ ∪ 𝑥 ∈ 𝐴 𝐷 ↔ ∃ 𝑥 ∈ 𝐴 𝑢 ∈ 𝐷 ) |
43 |
36 41 42
|
3bitr4g |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ( 𝑢 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑢 ∈ ∪ 𝑥 ∈ 𝐴 𝐷 ) ) |
44 |
43
|
eqrdv |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐷 ) |
45 |
|
df-fn |
⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 Fn ∪ 𝑥 ∈ 𝐴 𝐷 ↔ ( Fun ∪ 𝑥 ∈ 𝐴 𝐵 ∧ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐷 ) ) |
46 |
30 44 45
|
sylanbrc |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ∪ 𝑥 ∈ 𝐴 𝐵 Fn ∪ 𝑥 ∈ 𝐴 𝐷 ) |
47 |
|
rniun |
⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵 |
48 |
|
frn |
⊢ ( 𝐵 : 𝐷 ⟶ 𝑆 → ran 𝐵 ⊆ 𝑆 ) |
49 |
48
|
adantr |
⊢ ( ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ran 𝐵 ⊆ 𝑆 ) |
50 |
49
|
ralimi |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ∀ 𝑥 ∈ 𝐴 ran 𝐵 ⊆ 𝑆 ) |
51 |
|
iunss |
⊢ ( ∪ 𝑥 ∈ 𝐴 ran 𝐵 ⊆ 𝑆 ↔ ∀ 𝑥 ∈ 𝐴 ran 𝐵 ⊆ 𝑆 ) |
52 |
50 51
|
sylibr |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ∪ 𝑥 ∈ 𝐴 ran 𝐵 ⊆ 𝑆 ) |
53 |
47 52
|
eqsstrid |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ran ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑆 ) |
54 |
|
df-f |
⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 : ∪ 𝑥 ∈ 𝐴 𝐷 ⟶ 𝑆 ↔ ( ∪ 𝑥 ∈ 𝐴 𝐵 Fn ∪ 𝑥 ∈ 𝐴 𝐷 ∧ ran ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑆 ) ) |
55 |
46 53 54
|
sylanbrc |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 : 𝐷 ⟶ 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) → ∪ 𝑥 ∈ 𝐴 𝐵 : ∪ 𝑥 ∈ 𝐴 𝐷 ⟶ 𝑆 ) |