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