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

Theorem fncnv 5419
Description: Single-rootedness (see funcnv 5415) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
Assertion
Ref Expression
fncnv  |-  ( `' ( R  i^i  ( A  X.  B ) )  Fn  B  <->  A. y  e.  B  E! x  e.  A  x R
y )
Distinct variable groups:    x, y, A    x, B, y    x, R, y

Proof of Theorem fncnv
StepHypRef Expression
1 df-fn 5361 . 2  |-  ( `' ( R  i^i  ( A  X.  B ) )  Fn  B  <->  ( Fun  `' ( R  i^i  ( A  X.  B ) )  /\  dom  `' ( R  i^i  ( A  X.  B ) )  =  B ) )
2 df-rn 4803 . . . 4  |-  ran  ( R  i^i  ( A  X.  B ) )  =  dom  `' ( R  i^i  ( A  X.  B ) )
32eqeq1i 2373 . . 3  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  dom  `' ( R  i^i  ( A  X.  B ) )  =  B )
43anbi2i 675 . 2  |-  ( ( Fun  `' ( R  i^i  ( A  X.  B ) )  /\  ran  ( R  i^i  ( A  X.  B ) )  =  B )  <->  ( Fun  `' ( R  i^i  ( A  X.  B ) )  /\  dom  `' ( R  i^i  ( A  X.  B ) )  =  B ) )
5 rninxp 5220 . . . . 5  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  A. y  e.  B  E. x  e.  A  x R
y )
65anbi1i 676 . . . 4  |-  ( ( ran  ( R  i^i  ( A  X.  B
) )  =  B  /\  A. y  e.  B  E* x  e.  A x R y )  <->  ( A. y  e.  B  E. x  e.  A  x R
y  /\  A. y  e.  B  E* x  e.  A x R y ) )
7 funcnv 5415 . . . . . 6  |-  ( Fun  `' ( R  i^i  ( A  X.  B
) )  <->  A. y  e.  ran  ( R  i^i  ( A  X.  B
) ) E* x  x ( R  i^i  ( A  X.  B
) ) y )
8 raleq 2821 . . . . . . 7  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  ->  ( A. y  e.  ran  ( R  i^i  ( A  X.  B ) ) E* x  x ( R  i^i  ( A  X.  B ) ) y  <->  A. y  e.  B  E* x  x ( R  i^i  ( A  X.  B ) ) y ) )
9 biimt 325 . . . . . . . . 9  |-  ( y  e.  B  ->  ( E* x  e.  A x R y  <->  ( y  e.  B  ->  E* x  e.  A x R y ) ) )
10 moanimv 2275 . . . . . . . . . 10  |-  ( E* x ( y  e.  B  /\  ( x  e.  A  /\  x R y ) )  <-> 
( y  e.  B  ->  E* x ( x  e.  A  /\  x R y ) ) )
11 brinxp2 4854 . . . . . . . . . . . 12  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( x  e.  A  /\  y  e.  B  /\  x R y ) )
12 3anan12 948 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  y  e.  B  /\  x R y )  <->  ( y  e.  B  /\  (
x  e.  A  /\  x R y ) ) )
1311, 12bitri 240 . . . . . . . . . . 11  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( y  e.  B  /\  ( x  e.  A  /\  x R y ) ) )
1413mobii 2253 . . . . . . . . . 10  |-  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <->  E* x ( y  e.  B  /\  ( x  e.  A  /\  x R y ) ) )
15 df-rmo 2636 . . . . . . . . . . 11  |-  ( E* x  e.  A x R y  <->  E* x
( x  e.  A  /\  x R y ) )
1615imbi2i 303 . . . . . . . . . 10  |-  ( ( y  e.  B  ->  E* x  e.  A x R y )  <->  ( y  e.  B  ->  E* x
( x  e.  A  /\  x R y ) ) )
1710, 14, 163bitr4i 268 . . . . . . . . 9  |-  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <-> 
( y  e.  B  ->  E* x  e.  A x R y ) )
189, 17syl6rbbr 255 . . . . . . . 8  |-  ( y  e.  B  ->  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <->  E* x  e.  A x R y ) )
1918ralbiia 2660 . . . . . . 7  |-  ( A. y  e.  B  E* x  x ( R  i^i  ( A  X.  B
) ) y  <->  A. y  e.  B  E* x  e.  A x R y )
208, 19syl6bb 252 . . . . . 6  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  ->  ( A. y  e.  ran  ( R  i^i  ( A  X.  B ) ) E* x  x ( R  i^i  ( A  X.  B ) ) y  <->  A. y  e.  B  E* x  e.  A x R y ) )
217, 20syl5bb 248 . . . . 5  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  ->  ( Fun  `' ( R  i^i  ( A  X.  B
) )  <->  A. y  e.  B  E* x  e.  A x R y ) )
2221pm5.32i 618 . . . 4  |-  ( ( ran  ( R  i^i  ( A  X.  B
) )  =  B  /\  Fun  `' ( R  i^i  ( A  X.  B ) ) )  <->  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  /\  A. y  e.  B  E* x  e.  A x R y ) )
23 r19.26 2760 . . . 4  |-  ( A. y  e.  B  ( E. x  e.  A  x R y  /\  E* x  e.  A x R y )  <->  ( A. y  e.  B  E. x  e.  A  x R y  /\  A. y  e.  B  E* x  e.  A x R y ) )
246, 22, 233bitr4i 268 . . 3  |-  ( ( ran  ( R  i^i  ( A  X.  B
) )  =  B  /\  Fun  `' ( R  i^i  ( A  X.  B ) ) )  <->  A. y  e.  B  ( E. x  e.  A  x R y  /\  E* x  e.  A x R y ) )
25 ancom 437 . . 3  |-  ( ( Fun  `' ( R  i^i  ( A  X.  B ) )  /\  ran  ( R  i^i  ( A  X.  B ) )  =  B )  <->  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  /\  Fun  `' ( R  i^i  ( A  X.  B ) ) ) )
26 reu5 2838 . . . 4  |-  ( E! x  e.  A  x R y  <->  ( E. x  e.  A  x R y  /\  E* x  e.  A x R y ) )
2726ralbii 2652 . . 3  |-  ( A. y  e.  B  E! x  e.  A  x R y  <->  A. y  e.  B  ( E. x  e.  A  x R y  /\  E* x  e.  A x R y ) )
2824, 25, 273bitr4i 268 . 2  |-  ( ( Fun  `' ( R  i^i  ( A  X.  B ) )  /\  ran  ( R  i^i  ( A  X.  B ) )  =  B )  <->  A. y  e.  B  E! x  e.  A  x R
y )
291, 4, 283bitr2i 264 1  |-  ( `' ( R  i^i  ( A  X.  B ) )  Fn  B  <->  A. y  e.  B  E! x  e.  A  x R
y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   E*wmo 2218   A.wral 2628   E.wrex 2629   E!wreu 2630   E*wrmo 2631    i^i cin 3237   class class class wbr 4125    X. cxp 4790   `'ccnv 4791   dom cdm 4792   ran crn 4793   Fun wfun 5352    Fn wfn 5353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-fun 5360  df-fn 5361
  Copyright terms: Public domain W3C validator