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Theorem funcnvuni 4993
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 4985 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
funcnvuni  |-  ( A. f  e.  A  ( Fun  `' f  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  `' U. A )
Distinct variable group:    f, g, A

Proof of Theorem funcnvuni
Dummy variables  x  y  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnveq 4531 . . . . . . . 8  |-  ( x  =  v  ->  `' x  =  `' v
)
21eqeq2d 2093 . . . . . . 7  |-  ( x  =  v  ->  (
z  =  `' x  <->  z  =  `' v ) )
32cbvrexv 2579 . . . . . 6  |-  ( E. x  e.  A  z  =  `' x  <->  E. v  e.  A  z  =  `' v )
4 cnveq 4531 . . . . . . . . . . 11  |-  ( f  =  v  ->  `' f  =  `' v
)
54funeqd 4947 . . . . . . . . . 10  |-  ( f  =  v  ->  ( Fun  `' f  <->  Fun  `' v ) )
6 sseq1 3021 . . . . . . . . . . . 12  |-  ( f  =  v  ->  (
f  C_  g  <->  v  C_  g ) )
7 sseq2 3022 . . . . . . . . . . . 12  |-  ( f  =  v  ->  (
g  C_  f  <->  g  C_  v ) )
86, 7orbi12d 740 . . . . . . . . . . 11  |-  ( f  =  v  ->  (
( f  C_  g  \/  g  C_  f )  <-> 
( v  C_  g  \/  g  C_  v ) ) )
98ralbidv 2369 . . . . . . . . . 10  |-  ( f  =  v  ->  ( A. g  e.  A  ( f  C_  g  \/  g  C_  f )  <->  A. g  e.  A  ( v  C_  g  \/  g  C_  v ) ) )
105, 9anbi12d 457 . . . . . . . . 9  |-  ( 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 ) ) ) )
1110rspcv 2698 . . . . . . . 8  |-  ( 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 4945 . . . . . . . . . 10  |-  ( z  =  `' v  -> 
( Fun  z  <->  Fun  `' v ) )
1312biimprcd 158 . . . . . . . . 9  |-  ( Fun  `' v  ->  ( z  =  `' v  ->  Fun  z ) )
14 sseq2 3022 . . . . . . . . . . . . . . 15  |-  ( g  =  x  ->  (
v  C_  g  <->  v  C_  x ) )
15 sseq1 3021 . . . . . . . . . . . . . . 15  |-  ( g  =  x  ->  (
g  C_  v  <->  x  C_  v
) )
1614, 15orbi12d 740 . . . . . . . . . . . . . 14  |-  ( g  =  x  ->  (
( v  C_  g  \/  g  C_  v )  <-> 
( v  C_  x  \/  x  C_  v ) ) )
1716rspcv 2698 . . . . . . . . . . . . 13  |-  ( x  e.  A  ->  ( A. g  e.  A  ( v  C_  g  \/  g  C_  v )  ->  ( v  C_  x  \/  x  C_  v
) ) )
18 cnvss 4530 . . . . . . . . . . . . . . . 16  |-  ( v 
C_  x  ->  `' v  C_  `' x )
19 cnvss 4530 . . . . . . . . . . . . . . . 16  |-  ( x 
C_  v  ->  `' x  C_  `' v )
2018, 19orim12i 709 . . . . . . . . . . . . . . 15  |-  ( ( v  C_  x  \/  x  C_  v )  -> 
( `' v  C_  `' x  \/  `' x  C_  `' v ) )
21 sseq12 3023 . . . . . . . . . . . . . . . . 17  |-  ( ( z  =  `' v  /\  w  =  `' x )  ->  (
z  C_  w  <->  `' v  C_  `' x ) )
2221ancoms 264 . . . . . . . . . . . . . . . 16  |-  ( ( w  =  `' x  /\  z  =  `' v )  ->  (
z  C_  w  <->  `' v  C_  `' x ) )
23 sseq12 3023 . . . . . . . . . . . . . . . 16  |-  ( ( w  =  `' x  /\  z  =  `' v )  ->  (
w  C_  z  <->  `' x  C_  `' v ) )
2422, 23orbi12d 740 . . . . . . . . . . . . . . 15  |-  ( ( w  =  `' x  /\  z  =  `' v )  ->  (
( z  C_  w  \/  w  C_  z )  <-> 
( `' v  C_  `' x  \/  `' x  C_  `' v ) ) )
2520, 24syl5ibrcom 155 . . . . . . . . . . . . . 14  |-  ( ( v  C_  x  \/  x  C_  v )  -> 
( ( w  =  `' x  /\  z  =  `' v )  -> 
( z  C_  w  \/  w  C_  z ) ) )
2625expd 254 . . . . . . . . . . . . 13  |-  ( ( v  C_  x  \/  x  C_  v )  -> 
( w  =  `' x  ->  ( z  =  `' v  ->  ( z 
C_  w  \/  w  C_  z ) ) ) )
2717, 26syl6com 35 . . . . . . . . . . . 12  |-  ( A. g  e.  A  (
v  C_  g  \/  g  C_  v )  -> 
( x  e.  A  ->  ( w  =  `' x  ->  ( z  =  `' v  ->  ( z 
C_  w  \/  w  C_  z ) ) ) ) )
2827rexlimdv 2477 . . . . . . . . . . 11  |-  ( A. g  e.  A  (
v  C_  g  \/  g  C_  v )  -> 
( E. x  e.  A  w  =  `' x  ->  ( z  =  `' v  ->  ( z 
C_  w  \/  w  C_  z ) ) ) )
2928com23 77 . . . . . . . . . 10  |-  ( A. g  e.  A  (
v  C_  g  \/  g  C_  v )  -> 
( z  =  `' v  ->  ( E. x  e.  A  w  =  `' x  ->  ( z 
C_  w  \/  w  C_  z ) ) ) )
3029alrimdv 1798 . . . . . . . . 9  |-  ( 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 ) ) ) )
3113, 30anim12ii 335 . . . . . . . 8  |-  ( ( 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
) ) ) ) )
3211, 31syl6com 35 . . . . . . 7  |-  ( 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 ) ) ) ) ) )
3332rexlimdv 2477 . . . . . 6  |-  ( 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 ) ) ) ) )
343, 33syl5bi 150 . . . . 5  |-  ( 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 ) ) ) ) )
3534alrimiv 1796 . . . 4  |-  ( 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 2354 . . . . 5  |-  ( 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 2605 . . . . . . . 8  |-  z  e. 
_V
38 eqeq1 2088 . . . . . . . . 9  |-  ( y  =  z  ->  (
y  =  `' x  <->  z  =  `' x ) )
3938rexbidv 2370 . . . . . . . 8  |-  ( y  =  z  ->  ( E. x  e.  A  y  =  `' x  <->  E. x  e.  A  z  =  `' x ) )
4037, 39elab 2739 . . . . . . 7  |-  ( z  e.  { y  |  E. x  e.  A  y  =  `' x } 
<->  E. x  e.  A  z  =  `' x
)
41 eqeq1 2088 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y  =  `' x  <->  w  =  `' x ) )
4241rexbidv 2370 . . . . . . . . 9  |-  ( y  =  w  ->  ( E. x  e.  A  y  =  `' x  <->  E. x  e.  A  w  =  `' x ) )
4342ralab 2753 . . . . . . . 8  |-  ( 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 ) ) )
4443anbi2i 445 . . . . . . 7  |-  ( ( 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 ) ) ) )
4540, 44imbi12i 237 . . . . . 6  |-  ( ( 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 ) ) ) ) )
4645albii 1400 . . . . 5  |-  ( 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
) ) ) ) )
4736, 46bitr2i 183 . . . 4  |-  ( 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 ) ) )
4835, 47sylib 120 . . 3  |-  ( 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 4992 . . 3  |-  ( 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 } )
5048, 49syl 14 . 2  |-  ( 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 4543 . . . 4  |-  `' U. A  =  U_ x  e.  A  `' x
52 vex 2605 . . . . . 6  |-  x  e. 
_V
5352cnvex 4880 . . . . 5  |-  `' x  e.  _V
5453dfiun2 3714 . . . 4  |-  U_ x  e.  A  `' x  =  U. { y  |  E. x  e.  A  y  =  `' x }
5551, 54eqtri 2102 . . 3  |-  `' U. A  =  U. { y  |  E. x  e.  A  y  =  `' x }
5655funeqi 4946 . 2  |-  ( Fun  `' U. A  <->  Fun  U. {
y  |  E. x  e.  A  y  =  `' x } )
5750, 56sylibr 132 1  |-  ( A. f  e.  A  ( Fun  `' f  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  `' U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662   A.wal 1283    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350    C_ wss 2974   U.cuni 3603   U_ciun 3680   `'ccnv 4364   Fun wfun 4920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-iun 3682  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-fun 4928
This theorem is referenced by:  fun11uni  4994
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