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Theorem fncnv 4993
Description: Single-rootedness (see funcnv 4988) 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 4933 . 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 4384 . . . 4  |-  ran  ( R  i^i  ( A  X.  B ) )  =  dom  `' ( R  i^i  ( A  X.  B ) )
32eqeq1i 2063 . . 3  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  dom  `' ( R  i^i  ( A  X.  B ) )  =  B )
43anbi2i 438 . 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 4792 . . . . 5  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  A. y  e.  B  E. x  e.  A  x R
y )
65anbi1i 439 . . . 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 4988 . . . . . 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 2522 . . . . . . 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 234 . . . . . . . . 9  |-  ( y  e.  B  ->  ( E* x  e.  A  x R y  <->  ( y  e.  B  ->  E* x  e.  A  x R
y ) ) )
10 moanimv 1991 . . . . . . . . . 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 4435 . . . . . . . . . . . 12  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( x  e.  A  /\  y  e.  B  /\  x R y ) )
12 3anan12 908 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  y  e.  B  /\  x R y )  <->  ( y  e.  B  /\  (
x  e.  A  /\  x R y ) ) )
1311, 12bitri 177 . . . . . . . . . . 11  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( y  e.  B  /\  ( x  e.  A  /\  x R y ) ) )
1413mobii 1953 . . . . . . . . . 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 2331 . . . . . . . . . . 11  |-  ( E* x  e.  A  x R y  <->  E* x
( x  e.  A  /\  x R y ) )
1615imbi2i 219 . . . . . . . . . 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 205 . . . . . . . . 9  |-  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <-> 
( y  e.  B  ->  E* x  e.  A  x R y ) )
189, 17syl6rbbr 192 . . . . . . . 8  |-  ( y  e.  B  ->  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <->  E* x  e.  A  x R y ) )
1918ralbiia 2355 . . . . . . 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 189 . . . . . 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 185 . . . . 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 435 . . . 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 2458 . . . 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 205 . . 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 257 . . 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 2539 . . . 4  |-  ( E! x  e.  A  x R y  <->  ( E. x  e.  A  x R y  /\  E* x  e.  A  x R y ) )
2726ralbii 2347 . . 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 205 . 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 201 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    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   E*wmo 1917   A.wral 2323   E.wrex 2324   E!wreu 2325   E*wrmo 2326    i^i cin 2944   class class class wbr 3792    X. cxp 4371   `'ccnv 4372   dom cdm 4373   ran crn 4374   Fun wfun 4924    Fn wfn 4925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-fun 4932  df-fn 4933
This theorem is referenced by: (None)
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