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Theorem funcnvuni 6738
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5638 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 5166 . . . . . . . 8  |-  ( x  =  v  ->  `' x  =  `' v
)
21eqeq2d 2457 . . . . . . 7  |-  ( x  =  v  ->  (
z  =  `' x  <->  z  =  `' v ) )
32cbvrexv 3071 . . . . . 6  |-  ( E. x  e.  A  z  =  `' x  <->  E. v  e.  A  z  =  `' v )
4 cnveq 5166 . . . . . . . . . . 11  |-  ( f  =  v  ->  `' f  =  `' v
)
54funeqd 5599 . . . . . . . . . 10  |-  ( f  =  v  ->  ( Fun  `' f  <->  Fun  `' v ) )
6 sseq1 3510 . . . . . . . . . . . 12  |-  ( f  =  v  ->  (
f  C_  g  <->  v  C_  g ) )
7 sseq2 3511 . . . . . . . . . . . 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 2882 . . . . . . . . . 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 3192 . . . . . . . 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 5597 . . . . . . . . . 10  |-  ( z  =  `' v  -> 
( Fun  z  <->  Fun  `' v ) )
1312biimprcd 225 . . . . . . . . 9  |-  ( Fun  `' v  ->  ( z  =  `' v  ->  Fun  z ) )
14 sseq2 3511 . . . . . . . . . . . . . . 15  |-  ( g  =  x  ->  (
v  C_  g  <->  v  C_  x ) )
15 sseq1 3510 . . . . . . . . . . . . . . 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 3192 . . . . . . . . . . . . 13  |-  ( x  e.  A  ->  ( A. g  e.  A  ( v  C_  g  \/  g  C_  v )  ->  ( v  C_  x  \/  x  C_  v
) ) )
18 cnvss 5165 . . . . . . . . . . . . . . . 16  |-  ( v 
C_  x  ->  `' v  C_  `' x )
19 cnvss 5165 . . . . . . . . . . . . . . . 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 3512 . . . . . . . . . . . . . . . . 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 3512 . . . . . . . . . . . . . . . 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 2933 . . . . . . . . . . 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 1708 . . . . . . . . 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 2933 . . . . . 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 1706 . . . 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 2798 . . . . 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 3098 . . . . . . . 8  |-  z  e. 
_V
38 eqeq1 2447 . . . . . . . . 9  |-  ( y  =  z  ->  (
y  =  `' x  <->  z  =  `' x ) )
3938rexbidv 2954 . . . . . . . 8  |-  ( y  =  z  ->  ( E. x  e.  A  y  =  `' x  <->  E. x  e.  A  z  =  `' x ) )
4037, 39elab 3232 . . . . . . 7  |-  ( z  e.  { y  |  E. x  e.  A  y  =  `' x } 
<->  E. x  e.  A  z  =  `' x
)
41 eqeq1 2447 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y  =  `' x  <->  w  =  `' x ) )
4241rexbidv 2954 . . . . . . . . 9  |-  ( y  =  w  ->  ( E. x  e.  A  y  =  `' x  <->  E. x  e.  A  w  =  `' x ) )
4342ralab 3246 . . . . . . . 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 1627 . . . . 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 5644 . . 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 5179 . . . 4  |-  `' U. A  =  U_ x  e.  A  `' x
52 vex 3098 . . . . . 6  |-  x  e. 
_V
5352cnvex 6732 . . . . 5  |-  `' x  e.  _V
5453dfiun2 4349 . . . 4  |-  U_ x  e.  A  `' x  =  U. { y  |  E. x  e.  A  y  =  `' x }
5551, 54eqtri 2472 . . 3  |-  `' U. A  =  U. { y  |  E. x  e.  A  y  =  `' x }
5655funeqi 5598 . 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 1381    = wceq 1383    e. wcel 1804   {cab 2428   A.wral 2793   E.wrex 2794    C_ wss 3461   U.cuni 4234   U_ciun 4315   `'ccnv 4988   Fun wfun 5572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-fun 5580
This theorem is referenced by:  fun11uni  6739
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