| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ssmapsn.f |
⊢ Ⅎ 𝑓 𝐷 |
| 2 |
|
ssmapsn.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
| 3 |
|
ssmapsn.c |
⊢ ( 𝜑 → 𝐶 ⊆ ( 𝐵 ↑m { 𝐴 } ) ) |
| 4 |
|
ssmapsn.d |
⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓 |
| 5 |
3
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → 𝑓 ∈ ( 𝐵 ↑m { 𝐴 } ) ) |
| 6 |
|
elmapi |
⊢ ( 𝑓 ∈ ( 𝐵 ↑m { 𝐴 } ) → 𝑓 : { 𝐴 } ⟶ 𝐵 ) |
| 7 |
5 6
|
syl |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → 𝑓 : { 𝐴 } ⟶ 𝐵 ) |
| 8 |
7
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → 𝑓 Fn { 𝐴 } ) |
| 9 |
4
|
a1i |
⊢ ( 𝜑 → 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓 ) |
| 10 |
|
ovexd |
⊢ ( 𝜑 → ( 𝐵 ↑m { 𝐴 } ) ∈ V ) |
| 11 |
10 3
|
ssexd |
⊢ ( 𝜑 → 𝐶 ∈ V ) |
| 12 |
|
rnexg |
⊢ ( 𝑓 ∈ 𝐶 → ran 𝑓 ∈ V ) |
| 13 |
12
|
rgen |
⊢ ∀ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V |
| 14 |
|
iunexg |
⊢ ( ( 𝐶 ∈ V ∧ ∀ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V ) → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V ) |
| 15 |
11 13 14
|
sylancl |
⊢ ( 𝜑 → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V ) |
| 16 |
9 15
|
eqeltrd |
⊢ ( 𝜑 → 𝐷 ∈ V ) |
| 17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → 𝐷 ∈ V ) |
| 18 |
|
ssiun2 |
⊢ ( 𝑓 ∈ 𝐶 → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓 ) |
| 19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓 ) |
| 20 |
|
snidg |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 } ) |
| 21 |
2 20
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ { 𝐴 } ) |
| 22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → 𝐴 ∈ { 𝐴 } ) |
| 23 |
8 22
|
fnfvelrnd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑓 ) |
| 24 |
19 23
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → ( 𝑓 ‘ 𝐴 ) ∈ ∪ 𝑓 ∈ 𝐶 ran 𝑓 ) |
| 25 |
24 4
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → ( 𝑓 ‘ 𝐴 ) ∈ 𝐷 ) |
| 26 |
8 17 25
|
elmapsnd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐶 ) → 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) |
| 27 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) → 𝐷 ∈ V ) |
| 28 |
|
snex |
⊢ { 𝐴 } ∈ V |
| 29 |
28
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) → { 𝐴 } ∈ V ) |
| 30 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) → 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) |
| 31 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) → 𝐴 ∈ { 𝐴 } ) |
| 32 |
27 29 30 31
|
fvmap |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) → ( 𝑓 ‘ 𝐴 ) ∈ 𝐷 ) |
| 33 |
|
rneq |
⊢ ( 𝑓 = 𝑔 → ran 𝑓 = ran 𝑔 ) |
| 34 |
33
|
cbviunv |
⊢ ∪ 𝑓 ∈ 𝐶 ran 𝑓 = ∪ 𝑔 ∈ 𝐶 ran 𝑔 |
| 35 |
4 34
|
eqtri |
⊢ 𝐷 = ∪ 𝑔 ∈ 𝐶 ran 𝑔 |
| 36 |
32 35
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) → ( 𝑓 ‘ 𝐴 ) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔 ) |
| 37 |
|
eliun |
⊢ ( ( 𝑓 ‘ 𝐴 ) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔 ↔ ∃ 𝑔 ∈ 𝐶 ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) |
| 38 |
36 37
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) → ∃ 𝑔 ∈ 𝐶 ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) |
| 39 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) |
| 40 |
|
simp1l |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → 𝜑 ) |
| 41 |
40 2
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → 𝐴 ∈ 𝑉 ) |
| 42 |
|
eqid |
⊢ { 𝐴 } = { 𝐴 } |
| 43 |
|
simp1r |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) |
| 44 |
|
elmapfn |
⊢ ( 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) → 𝑓 Fn { 𝐴 } ) |
| 45 |
43 44
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → 𝑓 Fn { 𝐴 } ) |
| 46 |
3
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐶 ) → 𝑔 ∈ ( 𝐵 ↑m { 𝐴 } ) ) |
| 47 |
|
elmapfn |
⊢ ( 𝑔 ∈ ( 𝐵 ↑m { 𝐴 } ) → 𝑔 Fn { 𝐴 } ) |
| 48 |
46 47
|
syl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐶 ) → 𝑔 Fn { 𝐴 } ) |
| 49 |
48
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → 𝑔 Fn { 𝐴 } ) |
| 50 |
49
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → 𝑔 Fn { 𝐴 } ) |
| 51 |
41 42 45 50
|
fsneqrn |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → ( 𝑓 = 𝑔 ↔ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) ) |
| 52 |
39 51
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → 𝑓 = 𝑔 ) |
| 53 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → 𝑔 ∈ 𝐶 ) |
| 54 |
52 53
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ∧ 𝑔 ∈ 𝐶 ∧ ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 ) → 𝑓 ∈ 𝐶 ) |
| 55 |
54
|
rexlimdv3a |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) → ( ∃ 𝑔 ∈ 𝐶 ( 𝑓 ‘ 𝐴 ) ∈ ran 𝑔 → 𝑓 ∈ 𝐶 ) ) |
| 56 |
38 55
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) → 𝑓 ∈ 𝐶 ) |
| 57 |
26 56
|
impbida |
⊢ ( 𝜑 → ( 𝑓 ∈ 𝐶 ↔ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ) |
| 58 |
57
|
alrimiv |
⊢ ( 𝜑 → ∀ 𝑓 ( 𝑓 ∈ 𝐶 ↔ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ) |
| 59 |
|
nfcv |
⊢ Ⅎ 𝑓 𝐶 |
| 60 |
|
nfcv |
⊢ Ⅎ 𝑓 ↑m |
| 61 |
|
nfcv |
⊢ Ⅎ 𝑓 { 𝐴 } |
| 62 |
1 60 61
|
nfov |
⊢ Ⅎ 𝑓 ( 𝐷 ↑m { 𝐴 } ) |
| 63 |
59 62
|
cleqf |
⊢ ( 𝐶 = ( 𝐷 ↑m { 𝐴 } ) ↔ ∀ 𝑓 ( 𝑓 ∈ 𝐶 ↔ 𝑓 ∈ ( 𝐷 ↑m { 𝐴 } ) ) ) |
| 64 |
58 63
|
sylibr |
⊢ ( 𝜑 → 𝐶 = ( 𝐷 ↑m { 𝐴 } ) ) |