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Theorem funcnvuni 6727
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5639 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 5167 . . . . . . . 8  |-  ( x  =  v  ->  `' x  =  `' v
)
21eqeq2d 2474 . . . . . . 7  |-  ( x  =  v  ->  (
z  =  `' x  <->  z  =  `' v ) )
32cbvrexv 3082 . . . . . 6  |-  ( E. x  e.  A  z  =  `' x  <->  E. v  e.  A  z  =  `' v )
4 cnveq 5167 . . . . . . . . . . 11  |-  ( f  =  v  ->  `' f  =  `' v
)
54funeqd 5600 . . . . . . . . . 10  |-  ( f  =  v  ->  ( Fun  `' f  <->  Fun  `' v ) )
6 sseq1 3518 . . . . . . . . . . . 12  |-  ( f  =  v  ->  (
f  C_  g  <->  v  C_  g ) )
7 sseq2 3519 . . . . . . . . . . . 12  |-  ( f  =  v  ->  (
g  C_  f  <->  g  C_  v ) )
86, 7orbi12d 709 . . . . . . . . . . 11  |-  ( f  =  v  ->  (
( f  C_  g  \/  g  C_  f )  <-> 
( v  C_  g  \/  g  C_  v ) ) )
98ralbidv 2896 . . . . . . . . . 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 710 . . . . . . . . 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 3203 . . . . . . . 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 5598 . . . . . . . . . 10  |-  ( z  =  `' v  -> 
( Fun  z  <->  Fun  `' v ) )
1312biimprcd 225 . . . . . . . . 9  |-  ( Fun  `' v  ->  ( z  =  `' v  ->  Fun  z ) )
14 sseq2 3519 . . . . . . . . . . . . . . 15  |-  ( g  =  x  ->  (
v  C_  g  <->  v  C_  x ) )
15 sseq1 3518 . . . . . . . . . . . . . . 15  |-  ( g  =  x  ->  (
g  C_  v  <->  x  C_  v
) )
1614, 15orbi12d 709 . . . . . . . . . . . . . 14  |-  ( g  =  x  ->  (
( v  C_  g  \/  g  C_  v )  <-> 
( v  C_  x  \/  x  C_  v ) ) )
1716rspcv 3203 . . . . . . . . . . . . 13  |-  ( x  e.  A  ->  ( A. g  e.  A  ( v  C_  g  \/  g  C_  v )  ->  ( v  C_  x  \/  x  C_  v
) ) )
18 cnvss 5166 . . . . . . . . . . . . . . . 16  |-  ( v 
C_  x  ->  `' v  C_  `' x )
19 cnvss 5166 . . . . . . . . . . . . . . . 16  |-  ( x 
C_  v  ->  `' x  C_  `' v )
2018, 19orim12i 516 . . . . . . . . . . . . . . 15  |-  ( ( v  C_  x  \/  x  C_  v )  -> 
( `' v  C_  `' x  \/  `' x  C_  `' v ) )
21 sseq12 3520 . . . . . . . . . . . . . . . . 17  |-  ( ( z  =  `' v  /\  w  =  `' x )  ->  (
z  C_  w  <->  `' v  C_  `' x ) )
2221ancoms 453 . . . . . . . . . . . . . . . 16  |-  ( ( w  =  `' x  /\  z  =  `' v )  ->  (
z  C_  w  <->  `' v  C_  `' x ) )
23 sseq12 3520 . . . . . . . . . . . . . . . 16  |-  ( ( w  =  `' x  /\  z  =  `' v )  ->  (
w  C_  z  <->  `' x  C_  `' v ) )
2422, 23orbi12d 709 . . . . . . . . . . . . . . 15  |-  ( ( w  =  `' x  /\  z  =  `' v )  ->  (
( z  C_  w  \/  w  C_  z )  <-> 
( `' v  C_  `' x  \/  `' x  C_  `' v ) ) )
2520, 24syl5ibrcom 222 . . . . . . . . . . . . . 14  |-  ( ( v  C_  x  \/  x  C_  v )  -> 
( ( w  =  `' x  /\  z  =  `' v )  -> 
( z  C_  w  \/  w  C_  z ) ) )
2625expd 436 . . . . . . . . . . . . 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 2946 . . . . . . . . . . 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 78 . . . . . . . . . 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 1692 . . . . . . . . 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 570 . . . . . . . 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 2946 . . . . . 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 217 . . . . 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 1690 . . . 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 2812 . . . . 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 3109 . . . . . . . 8  |-  z  e. 
_V
38 eqeq1 2464 . . . . . . . . 9  |-  ( y  =  z  ->  (
y  =  `' x  <->  z  =  `' x ) )
3938rexbidv 2966 . . . . . . . 8  |-  ( y  =  z  ->  ( E. x  e.  A  y  =  `' x  <->  E. x  e.  A  z  =  `' x ) )
4037, 39elab 3243 . . . . . . 7  |-  ( z  e.  { y  |  E. x  e.  A  y  =  `' x } 
<->  E. x  e.  A  z  =  `' x
)
41 eqeq1 2464 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y  =  `' x  <->  w  =  `' x ) )
4241rexbidv 2966 . . . . . . . . 9  |-  ( y  =  w  ->  ( E. x  e.  A  y  =  `' x  <->  E. x  e.  A  w  =  `' x ) )
4342ralab 3257 . . . . . . . 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 694 . . . . . . 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 326 . . . . . 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 1615 . . . . 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 250 . . . 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 196 . . 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 5645 . . 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 16 . 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 5180 . . . 4  |-  `' U. A  =  U_ x  e.  A  `' x
52 vex 3109 . . . . . 6  |-  x  e. 
_V
5352cnvex 6721 . . . . 5  |-  `' x  e.  _V
5453dfiun2 4352 . . . 4  |-  U_ x  e.  A  `' x  =  U. { y  |  E. x  e.  A  y  =  `' x }
5551, 54eqtri 2489 . . 3  |-  `' U. A  =  U. { y  |  E. x  e.  A  y  =  `' x }
5655funeqi 5599 . 2  |-  ( Fun  `' U. A  <->  Fun  U. {
y  |  E. x  e.  A  y  =  `' x } )
5750, 56sylibr 212 1  |-  ( A. f  e.  A  ( Fun  `' f  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  `' U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1372    = wceq 1374    e. wcel 1762   {cab 2445   A.wral 2807   E.wrex 2808    C_ wss 3469   U.cuni 4238   U_ciun 4318   `'ccnv 4991   Fun wfun 5573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-fun 5581
This theorem is referenced by:  fun11uni  6728
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