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Theorem fncnv 5478
Description: Single-rootedness (see funcnv 5474) 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 5420 . 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 4852 . . . 4  |-  ran  ( R  i^i  ( A  X.  B ) )  =  dom  `' ( R  i^i  ( A  X.  B ) )
32eqeq1i 2415 . . 3  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  dom  `' ( R  i^i  ( A  X.  B ) )  =  B )
43anbi2i 676 . 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 5273 . . . . 5  |-  ( ran  ( R  i^i  ( A  X.  B ) )  =  B  <->  A. y  e.  B  E. x  e.  A  x R
y )
65anbi1i 677 . . . 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 5474 . . . . . 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 2868 . . . . . . 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 326 . . . . . . . . 9  |-  ( y  e.  B  ->  ( E* x  e.  A x R y  <->  ( y  e.  B  ->  E* x  e.  A x R y ) ) )
10 moanimv 2316 . . . . . . . . . 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 4902 . . . . . . . . . . . 12  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( x  e.  A  /\  y  e.  B  /\  x R y ) )
12 3anan12 949 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  y  e.  B  /\  x R y )  <->  ( y  e.  B  /\  (
x  e.  A  /\  x R y ) ) )
1311, 12bitri 241 . . . . . . . . . . 11  |-  ( x ( R  i^i  ( A  X.  B ) ) y  <->  ( y  e.  B  /\  ( x  e.  A  /\  x R y ) ) )
1413mobii 2294 . . . . . . . . . 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 2678 . . . . . . . . . . 11  |-  ( E* x  e.  A x R y  <->  E* x
( x  e.  A  /\  x R y ) )
1615imbi2i 304 . . . . . . . . . 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 269 . . . . . . . . 9  |-  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <-> 
( y  e.  B  ->  E* x  e.  A x R y ) )
189, 17syl6rbbr 256 . . . . . . . 8  |-  ( y  e.  B  ->  ( E* x  x ( R  i^i  ( A  X.  B ) ) y  <->  E* x  e.  A x R y ) )
1918ralbiia 2702 . . . . . . 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 253 . . . . . 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 249 . . . . 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 619 . . . 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 2802 . . . 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 269 . . 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 438 . . 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 2885 . . . 4  |-  ( E! x  e.  A  x R y  <->  ( E. x  e.  A  x R y  /\  E* x  e.  A x R y ) )
2726ralbii 2694 . . 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 269 . 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 265 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E*wmo 2259   A.wral 2670   E.wrex 2671   E!wreu 2672   E*wrmo 2673    i^i cin 3283   class class class wbr 4176    X. cxp 4839   `'ccnv 4840   dom cdm 4841   ran crn 4842   Fun wfun 5411    Fn wfn 5412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-fun 5419  df-fn 5420
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