Metamath Proof Explorer

Theorem funcnvuni

Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004)

Ref Expression
Assertion funcnvuni 
|- ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun `' U. A )

Proof

Step Hyp Ref Expression
1 cnveq 
 |-  ( x = v -> `' x = `' v )
2 1 eqeq2d 
 |-  ( x = v -> ( z = `' x <-> z = `' v ) )
3 2 cbvrexvw 
 |-  ( E. x e. A z = `' x <-> E. v e. A z = `' v )
4 cnveq 
 |-  ( f = v -> `' f = `' v )
5 4 funeqd 
 |-  ( f = v -> ( Fun `' f <-> Fun `' v ) )
6 sseq1 
 |-  ( f = v -> ( f C_ g <-> v C_ g ) )
7 sseq2 
 |-  ( f = v -> ( g C_ f <-> g C_ v ) )
8 6 7 orbi12d 
 |-  ( f = v -> ( ( f C_ g \/ g C_ f ) <-> ( v C_ g \/ g C_ v ) ) )
9 8 ralbidv 
 |-  ( f = v -> ( A. g e. A ( f C_ g \/ g C_ f ) <-> A. g e. A ( v C_ g \/ g C_ v ) ) )
10 5 9 anbi12d 
 |-  ( f = v -> ( ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) <-> ( Fun `' v /\ A. g e. A ( v C_ g \/ g C_ v ) ) ) )
11 10 rspcv 
 |-  ( v e. A -> ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( Fun `' v /\ A. g e. A ( v C_ g \/ g C_ v ) ) ) )
12 funeq 
 |-  ( z = `' v -> ( Fun z <-> Fun `' v ) )
13 12 biimprcd 
 |-  ( Fun `' v -> ( z = `' v -> Fun z ) )
14 sseq2 
 |-  ( g = x -> ( v C_ g <-> v C_ x ) )
15 sseq1 
 |-  ( g = x -> ( g C_ v <-> x C_ v ) )
16 14 15 orbi12d 
 |-  ( g = x -> ( ( v C_ g \/ g C_ v ) <-> ( v C_ x \/ x C_ v ) ) )
17 16 rspcv 
 |-  ( x e. A -> ( A. g e. A ( v C_ g \/ g C_ v ) -> ( v C_ x \/ x C_ v ) ) )
18 cnvss 
 |-  ( v C_ x -> `' v C_ `' x )
19 cnvss 
 |-  ( x C_ v -> `' x C_ `' v )
20 18 19 orim12i 
 |-  ( ( v C_ x \/ x C_ v ) -> ( `' v C_ `' x \/ `' x C_ `' v ) )
21 sseq12 
 |-  ( ( z = `' v /\ w = `' x ) -> ( z C_ w <-> `' v C_ `' x ) )
22 21 ancoms 
 |-  ( ( w = `' x /\ z = `' v ) -> ( z C_ w <-> `' v C_ `' x ) )
23 sseq12 
 |-  ( ( w = `' x /\ z = `' v ) -> ( w C_ z <-> `' x C_ `' v ) )
24 22 23 orbi12d 
 |-  ( ( w = `' x /\ z = `' v ) -> ( ( z C_ w \/ w C_ z ) <-> ( `' v C_ `' x \/ `' x C_ `' v ) ) )
25 20 24 syl5ibrcom 
 |-  ( ( v C_ x \/ x C_ v ) -> ( ( w = `' x /\ z = `' v ) -> ( z C_ w \/ w C_ z ) ) )
26 25 expd 
 |-  ( ( v C_ x \/ x C_ v ) -> ( w = `' x -> ( z = `' v -> ( z C_ w \/ w C_ z ) ) ) )
27 17 26 syl6com 
 |-  ( A. g e. A ( v C_ g \/ g C_ v ) -> ( x e. A -> ( w = `' x -> ( z = `' v -> ( z C_ w \/ w C_ z ) ) ) ) )
28 27 rexlimdv 
 |-  ( A. g e. A ( v C_ g \/ g C_ v ) -> ( E. x e. A w = `' x -> ( z = `' v -> ( z C_ w \/ w C_ z ) ) ) )
29 28 com23 
 |-  ( A. g e. A ( v C_ g \/ g C_ v ) -> ( z = `' v -> ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) )
30 29 alrimdv 
 |-  ( A. g e. A ( v C_ g \/ g C_ v ) -> ( z = `' v -> A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) )
31 13 30 anim12ii 
 |-  ( ( Fun `' v /\ A. g e. A ( v C_ g \/ g C_ v ) ) -> ( z = `' v -> ( Fun z /\ A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) ) )
32 11 31 syl6com 
 |-  ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( v e. A -> ( z = `' v -> ( Fun z /\ A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) ) ) )
33 32 rexlimdv 
 |-  ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( E. v e. A z = `' v -> ( Fun z /\ A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) ) )
34 3 33 syl5bi 
 |-  ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( E. x e. A z = `' x -> ( Fun z /\ A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) ) )
35 34 alrimiv 
 |-  ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> A. z ( E. x e. A z = `' x -> ( Fun z /\ A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) ) )
36 df-ral 
 |-  ( A. z e. { y | E. x e. A y = `' x } ( Fun z /\ A. w e. { y | E. x e. A y = `' x } ( z C_ w \/ w C_ z ) ) <-> A. z ( z e. { y | E. x e. A y = `' x } -> ( Fun z /\ A. w e. { y | E. x e. A y = `' x } ( z C_ w \/ w C_ z ) ) ) )
37 vex 
 |-  z e. _V
38 eqeq1 
 |-  ( y = z -> ( y = `' x <-> z = `' x ) )
39 38 rexbidv 
 |-  ( y = z -> ( E. x e. A y = `' x <-> E. x e. A z = `' x ) )
40 37 39 elab 
 |-  ( z e. { y | E. x e. A y = `' x } <-> E. x e. A z = `' x )
41 eqeq1 
 |-  ( y = w -> ( y = `' x <-> w = `' x ) )
42 41 rexbidv 
 |-  ( y = w -> ( E. x e. A y = `' x <-> E. x e. A w = `' x ) )
43 42 ralab 
 |-  ( A. w e. { y | E. x e. A y = `' x } ( z C_ w \/ w C_ z ) <-> A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) )
44 43 anbi2i 
 |-  ( ( Fun z /\ A. w e. { y | E. x e. A y = `' x } ( z C_ w \/ w C_ z ) ) <-> ( Fun z /\ A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) )
45 40 44 imbi12i 
 |-  ( ( z e. { y | E. x e. A y = `' x } -> ( Fun z /\ A. w e. { y | E. x e. A y = `' x } ( z C_ w \/ w C_ z ) ) ) <-> ( E. x e. A z = `' x -> ( Fun z /\ A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) ) )
46 45 albii 
 |-  ( A. z ( z e. { y | E. x e. A y = `' x } -> ( Fun z /\ A. w e. { y | E. x e. A y = `' x } ( z C_ w \/ w C_ z ) ) ) <-> A. z ( E. x e. A z = `' x -> ( Fun z /\ A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) ) )
47 36 46 bitr2i 
 |-  ( A. z ( E. x e. A z = `' x -> ( Fun z /\ A. w ( E. x e. A w = `' x -> ( z C_ w \/ w C_ z ) ) ) ) <-> A. z e. { y | E. x e. A y = `' x } ( Fun z /\ A. w e. { y | E. x e. A y = `' x } ( z C_ w \/ w C_ z ) ) )
48 35 47 sylib 
 |-  ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> A. z e. { y | E. x e. A y = `' x } ( Fun z /\ A. w e. { y | E. x e. A y = `' x } ( z C_ w \/ w C_ z ) ) )
49 fununi 
 |-  ( A. z e. { y | E. x e. A y = `' x } ( Fun z /\ A. w e. { y | E. x e. A y = `' x } ( z C_ w \/ w C_ z ) ) -> Fun U. { y | E. x e. A y = `' x } )
50 48 49 syl 
 |-  ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun U. { y | E. x e. A y = `' x } )
51 cnvuni 
 |-  `' U. A = U_ x e. A `' x
52 vex 
 |-  x e. _V
53 52 cnvex 
 |-  `' x e. _V
54 53 dfiun2 
 |-  U_ x e. A `' x = U. { y | E. x e. A y = `' x }
55 51 54 eqtri 
 |-  `' U. A = U. { y | E. x e. A y = `' x }
56 55 funeqi 
 |-  ( Fun `' U. A <-> Fun U. { y | E. x e. A y = `' x } )
57 50 56 sylibr 
 |-  ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun `' U. A )