MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcnvuni Unicode version

Theorem funcnvuni 5333
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5326 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 4871 . . . . . . . 8  |-  ( x  =  v  ->  `' x  =  `' v
)
21eqeq2d 2307 . . . . . . 7  |-  ( x  =  v  ->  (
z  =  `' x  <->  z  =  `' v ) )
32cbvrexv 2778 . . . . . 6  |-  ( E. x  e.  A  z  =  `' x  <->  E. v  e.  A  z  =  `' v )
4 cnveq 4871 . . . . . . . . . . 11  |-  ( f  =  v  ->  `' f  =  `' v
)
54funeqd 5292 . . . . . . . . . 10  |-  ( f  =  v  ->  ( Fun  `' f  <->  Fun  `' v ) )
6 sseq1 3212 . . . . . . . . . . . 12  |-  ( f  =  v  ->  (
f  C_  g  <->  v  C_  g ) )
7 sseq2 3213 . . . . . . . . . . . 12  |-  ( f  =  v  ->  (
g  C_  f  <->  g  C_  v ) )
86, 7orbi12d 690 . . . . . . . . . . 11  |-  ( f  =  v  ->  (
( f  C_  g  \/  g  C_  f )  <-> 
( v  C_  g  \/  g  C_  v ) ) )
98ralbidv 2576 . . . . . . . . . 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 691 . . . . . . . . 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 2893 . . . . . . . 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 5290 . . . . . . . . . 10  |-  ( z  =  `' v  -> 
( Fun  z  <->  Fun  `' v ) )
1312biimprcd 216 . . . . . . . . 9  |-  ( Fun  `' v  ->  ( z  =  `' v  ->  Fun  z ) )
14 sseq2 3213 . . . . . . . . . . . . . . 15  |-  ( g  =  x  ->  (
v  C_  g  <->  v  C_  x ) )
15 sseq1 3212 . . . . . . . . . . . . . . 15  |-  ( g  =  x  ->  (
g  C_  v  <->  x  C_  v
) )
1614, 15orbi12d 690 . . . . . . . . . . . . . 14  |-  ( g  =  x  ->  (
( v  C_  g  \/  g  C_  v )  <-> 
( v  C_  x  \/  x  C_  v ) ) )
1716rspcv 2893 . . . . . . . . . . . . 13  |-  ( x  e.  A  ->  ( A. g  e.  A  ( v  C_  g  \/  g  C_  v )  ->  ( v  C_  x  \/  x  C_  v
) ) )
18 cnvss 4870 . . . . . . . . . . . . . . . 16  |-  ( v 
C_  x  ->  `' v  C_  `' x )
19 cnvss 4870 . . . . . . . . . . . . . . . 16  |-  ( x 
C_  v  ->  `' x  C_  `' v )
2018, 19orim12i 502 . . . . . . . . . . . . . . 15  |-  ( ( v  C_  x  \/  x  C_  v )  -> 
( `' v  C_  `' x  \/  `' x  C_  `' v ) )
21 sseq12 3214 . . . . . . . . . . . . . . . . 17  |-  ( ( z  =  `' v  /\  w  =  `' x )  ->  (
z  C_  w  <->  `' v  C_  `' x ) )
2221ancoms 439 . . . . . . . . . . . . . . . 16  |-  ( ( w  =  `' x  /\  z  =  `' v )  ->  (
z  C_  w  <->  `' v  C_  `' x ) )
23 sseq12 3214 . . . . . . . . . . . . . . . 16  |-  ( ( w  =  `' x  /\  z  =  `' v )  ->  (
w  C_  z  <->  `' x  C_  `' v ) )
2422, 23orbi12d 690 . . . . . . . . . . . . . . 15  |-  ( ( w  =  `' x  /\  z  =  `' v )  ->  (
( z  C_  w  \/  w  C_  z )  <-> 
( `' v  C_  `' x  \/  `' x  C_  `' v ) ) )
2520, 24syl5ibrcom 213 . . . . . . . . . . . . . 14  |-  ( ( v  C_  x  \/  x  C_  v )  -> 
( ( w  =  `' x  /\  z  =  `' v )  -> 
( z  C_  w  \/  w  C_  z ) ) )
2625exp3a 425 . . . . . . . . . . . . 13  |-  ( ( v  C_  x  \/  x  C_  v )  -> 
( w  =  `' x  ->  ( z  =  `' v  ->  ( z 
C_  w  \/  w  C_  z ) ) ) )
2717, 26syl6com 31 . . . . . . . . . . . 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 2679 . . . . . . . . . . 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 72 . . . . . . . . . 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 1623 . . . . . . . . 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 553 . . . . . . . 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 31 . . . . . . 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 2679 . . . . . 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 208 . . . . 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 1621 . . . 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 2561 . . . . 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 2804 . . . . . . . 8  |-  z  e. 
_V
38 eqeq1 2302 . . . . . . . . 9  |-  ( y  =  z  ->  (
y  =  `' x  <->  z  =  `' x ) )
3938rexbidv 2577 . . . . . . . 8  |-  ( y  =  z  ->  ( E. x  e.  A  y  =  `' x  <->  E. x  e.  A  z  =  `' x ) )
4037, 39elab 2927 . . . . . . 7  |-  ( z  e.  { y  |  E. x  e.  A  y  =  `' x } 
<->  E. x  e.  A  z  =  `' x
)
41 eqeq1 2302 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y  =  `' x  <->  w  =  `' x ) )
4241rexbidv 2577 . . . . . . . . 9  |-  ( y  =  w  ->  ( E. x  e.  A  y  =  `' x  <->  E. x  e.  A  w  =  `' x ) )
4342ralab 2939 . . . . . . . 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 675 . . . . . . 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 316 . . . . . 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 1556 . . . . 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 241 . . . 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 188 . . 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 5332 . . 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 15 . 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 4882 . . . 4  |-  `' U. A  =  U_ x  e.  A  `' x
52 vex 2804 . . . . . 6  |-  x  e. 
_V
5352cnvex 5225 . . . . 5  |-  `' x  e.  _V
5453dfiun2 3953 . . . 4  |-  U_ x  e.  A  `' x  =  U. { y  |  E. x  e.  A  y  =  `' x }
5551, 54eqtri 2316 . . 3  |-  `' U. A  =  U. { y  |  E. x  e.  A  y  =  `' x }
5655funeqi 5291 . 2  |-  ( Fun  `' U. A  <->  Fun  U. {
y  |  E. x  e.  A  y  =  `' x } )
5750, 56sylibr 203 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    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    C_ wss 3165   U.cuni 3843   U_ciun 3921   `'ccnv 4704   Fun wfun 5265
This theorem is referenced by:  fun11uni  5334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273
  Copyright terms: Public domain W3C validator