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Theorem fncnv 5518
Description: Single-rootedness (see funcnv 5514) 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 5460 . 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 4892 . . . 4  |-  ran  ( R  i^i  ( A  X.  B ) )  =  dom  `' ( R  i^i  ( A  X.  B ) )
32eqeq1i 2445 . . 3  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  dom  `' ( R  i^i  ( A  X.  B ) )  =  B )
43anbi2i 677 . 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 5313 . . . . 5  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  A. y  e.  B  E. x  e.  A  x R
y )
65anbi1i 678 . . . 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 5514 . . . . . 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 2906 . . . . . . 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 327 . . . . . . . . 9  |-  ( y  e.  B  ->  ( E* x  e.  A x R y  <->  ( y  e.  B  ->  E* x  e.  A x R y ) ) )
10 moanimv 2341 . . . . . . . . . 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 4942 . . . . . . . . . . . 12  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( x  e.  A  /\  y  e.  B  /\  x R y ) )
12 3anan12 950 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  y  e.  B  /\  x R y )  <->  ( y  e.  B  /\  (
x  e.  A  /\  x R y ) ) )
1311, 12bitri 242 . . . . . . . . . . 11  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( y  e.  B  /\  ( x  e.  A  /\  x R y ) ) )
1413mobii 2319 . . . . . . . . . 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 2715 . . . . . . . . . . 11  |-  ( E* x  e.  A x R y  <->  E* x
( x  e.  A  /\  x R y ) )
1615imbi2i 305 . . . . . . . . . 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 270 . . . . . . . . 9  |-  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <-> 
( y  e.  B  ->  E* x  e.  A x R y ) )
189, 17syl6rbbr 257 . . . . . . . 8  |-  ( y  e.  B  ->  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <->  E* x  e.  A x R y ) )
1918ralbiia 2739 . . . . . . 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 254 . . . . . 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 250 . . . . 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 620 . . . 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 2840 . . . 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 270 . . 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 439 . . 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 2923 . . . 4  |-  ( E! x  e.  A  x R y  <->  ( E. x  e.  A  x R y  /\  E* x  e.  A x R y ) )
2726ralbii 2731 . . 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 270 . 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 266 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E*wmo 2284   A.wral 2707   E.wrex 2708   E!wreu 2709   E*wrmo 2710    i^i cin 3321   class class class wbr 4215    X. cxp 4879   `'ccnv 4880   dom cdm 4881   ran crn 4882   Fun wfun 5451    Fn wfn 5452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-fun 5459  df-fn 5460
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