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Theorem fncnv 5658
Description: Single-rootedness (see funcnv 5654) of a class cut down by a Cartesian 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 5597 . 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 5016 . . . 4  |-  ran  ( R  i^i  ( A  X.  B ) )  =  dom  `' ( R  i^i  ( A  X.  B ) )
32eqeq1i 2474 . . 3  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  dom  `' ( R  i^i  ( A  X.  B ) )  =  B )
43anbi2i 694 . 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 5452 . . . . 5  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  A. y  e.  B  E. x  e.  A  x R
y )
65anbi1i 695 . . . 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 5654 . . . . . 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 3063 . . . . . . 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 335 . . . . . . . . 9  |-  ( y  e.  B  ->  ( E* x  e.  A  x R y  <->  ( y  e.  B  ->  E* x  e.  A  x R
y ) ) )
10 moanimv 2356 . . . . . . . . . 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 5067 . . . . . . . . . . . 12  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( x  e.  A  /\  y  e.  B  /\  x R y ) )
12 3anan12 986 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  y  e.  B  /\  x R y )  <->  ( y  e.  B  /\  (
x  e.  A  /\  x R y ) ) )
1311, 12bitri 249 . . . . . . . . . . 11  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( y  e.  B  /\  ( x  e.  A  /\  x R y ) ) )
1413mobii 2301 . . . . . . . . . 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 2825 . . . . . . . . . . 11  |-  ( E* x  e.  A  x R y  <->  E* x
( x  e.  A  /\  x R y ) )
1615imbi2i 312 . . . . . . . . . 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 277 . . . . . . . . 9  |-  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <-> 
( y  e.  B  ->  E* x  e.  A  x R y ) )
189, 17syl6rbbr 264 . . . . . . . 8  |-  ( y  e.  B  ->  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <->  E* x  e.  A  x R y ) )
1918ralbiia 2897 . . . . . . 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 261 . . . . . 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 257 . . . . 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 637 . . . 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 2994 . . . 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 277 . . 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 450 . . 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 3082 . . . 4  |-  ( E! x  e.  A  x R y  <->  ( E. x  e.  A  x R y  /\  E* x  e.  A  x R y ) )
2726ralbii 2898 . . 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 277 . 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 273 1  |-  ( `' ( R  i^i  ( A  X.  B ) )  Fn  B  <->  A. y  e.  B  E! x  e.  A  x R
y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E*wmo 2276   A.wral 2817   E.wrex 2818   E!wreu 2819   E*wrmo 2820    i^i cin 3480   class class class wbr 4453    X. cxp 5003   `'ccnv 5004   dom cdm 5005   ran crn 5006   Fun wfun 5588    Fn wfn 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-fun 5596  df-fn 5597
This theorem is referenced by: (None)
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