Step |
Hyp |
Ref |
Expression |
1 |
|
actfunsn.1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) → 𝐴 ⊆ ( 𝐶 ↑m 𝐵 ) ) |
2 |
|
actfunsn.2 |
⊢ ( 𝜑 → 𝐶 ∈ V ) |
3 |
|
actfunsn.3 |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
4 |
|
actfunsn.4 |
⊢ ( 𝜑 → ¬ 𝐼 ∈ 𝐵 ) |
5 |
|
actfunsn.5 |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝑥 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) |
6 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑓 = ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) → 𝑓 = ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) |
7 |
6
|
fveq1d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑓 = ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) → ( 𝑓 ‘ 𝐼 ) = ( ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ‘ 𝐼 ) ) |
8 |
1
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → 𝐴 ⊆ ( 𝐶 ↑m 𝐵 ) ) |
9 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ 𝐴 ) |
10 |
8 9
|
sseldd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ ( 𝐶 ↑m 𝐵 ) ) |
11 |
|
elmapfn |
⊢ ( 𝑧 ∈ ( 𝐶 ↑m 𝐵 ) → 𝑧 Fn 𝐵 ) |
12 |
10 11
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → 𝑧 Fn 𝐵 ) |
13 |
3
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → 𝐼 ∈ 𝑉 ) |
14 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → 𝑘 ∈ 𝐶 ) |
15 |
|
fnsng |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶 ) → { 〈 𝐼 , 𝑘 〉 } Fn { 𝐼 } ) |
16 |
13 14 15
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → { 〈 𝐼 , 𝑘 〉 } Fn { 𝐼 } ) |
17 |
|
disjsn |
⊢ ( ( 𝐵 ∩ { 𝐼 } ) = ∅ ↔ ¬ 𝐼 ∈ 𝐵 ) |
18 |
4 17
|
sylibr |
⊢ ( 𝜑 → ( 𝐵 ∩ { 𝐼 } ) = ∅ ) |
19 |
18
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐵 ∩ { 𝐼 } ) = ∅ ) |
20 |
|
snidg |
⊢ ( 𝐼 ∈ 𝑉 → 𝐼 ∈ { 𝐼 } ) |
21 |
13 20
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → 𝐼 ∈ { 𝐼 } ) |
22 |
|
fvun2 |
⊢ ( ( 𝑧 Fn 𝐵 ∧ { 〈 𝐼 , 𝑘 〉 } Fn { 𝐼 } ∧ ( ( 𝐵 ∩ { 𝐼 } ) = ∅ ∧ 𝐼 ∈ { 𝐼 } ) ) → ( ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ‘ 𝐼 ) = ( { 〈 𝐼 , 𝑘 〉 } ‘ 𝐼 ) ) |
23 |
12 16 19 21 22
|
syl112anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ‘ 𝐼 ) = ( { 〈 𝐼 , 𝑘 〉 } ‘ 𝐼 ) ) |
24 |
|
fvsng |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶 ) → ( { 〈 𝐼 , 𝑘 〉 } ‘ 𝐼 ) = 𝑘 ) |
25 |
13 14 24
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → ( { 〈 𝐼 , 𝑘 〉 } ‘ 𝐼 ) = 𝑘 ) |
26 |
23 25
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ‘ 𝐼 ) = 𝑘 ) |
27 |
26
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑓 = ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) → ( ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ‘ 𝐼 ) = 𝑘 ) |
28 |
7 27
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑓 = ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) → ( 𝑓 ‘ 𝐼 ) = 𝑘 ) |
29 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) → 𝑓 ∈ ran 𝐹 ) |
30 |
|
uneq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∪ { 〈 𝐼 , 𝑘 〉 } ) = ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) |
31 |
30
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝐴 ↦ ( 𝑥 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) = ( 𝑧 ∈ 𝐴 ↦ ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) |
32 |
5 31
|
eqtri |
⊢ 𝐹 = ( 𝑧 ∈ 𝐴 ↦ ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) |
33 |
|
vex |
⊢ 𝑧 ∈ V |
34 |
|
snex |
⊢ { 〈 𝐼 , 𝑘 〉 } ∈ V |
35 |
33 34
|
unex |
⊢ ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ∈ V |
36 |
32 35
|
elrnmpti |
⊢ ( 𝑓 ∈ ran 𝐹 ↔ ∃ 𝑧 ∈ 𝐴 𝑓 = ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) |
37 |
29 36
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) → ∃ 𝑧 ∈ 𝐴 𝑓 = ( 𝑧 ∪ { 〈 𝐼 , 𝑘 〉 } ) ) |
38 |
28 37
|
r19.29a |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ∧ 𝑓 ∈ ran 𝐹 ) → ( 𝑓 ‘ 𝐼 ) = 𝑘 ) |
39 |
38
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) → ∀ 𝑓 ∈ ran 𝐹 ( 𝑓 ‘ 𝐼 ) = 𝑘 ) |
40 |
39
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐶 ∀ 𝑓 ∈ ran 𝐹 ( 𝑓 ‘ 𝐼 ) = 𝑘 ) |
41 |
|
invdisj |
⊢ ( ∀ 𝑘 ∈ 𝐶 ∀ 𝑓 ∈ ran 𝐹 ( 𝑓 ‘ 𝐼 ) = 𝑘 → Disj 𝑘 ∈ 𝐶 ran 𝐹 ) |
42 |
40 41
|
syl |
⊢ ( 𝜑 → Disj 𝑘 ∈ 𝐶 ran 𝐹 ) |