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Theorem f1ompt 5571
Description: Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)
Hypothesis
Ref Expression
fmpt.1  |-  F  =  ( x  e.  A  |->  C )
Assertion
Ref Expression
f1ompt  |-  ( F : A -1-1-onto-> B  <->  ( A. x  e.  A  C  e.  B  /\  A. y  e.  B  E! x  e.  A  y  =  C ) )
Distinct variable groups:    x, y, A   
x, B, y    y, C    y, F
Allowed substitution hints:    C( x)    F( x)

Proof of Theorem f1ompt
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ffn 5272 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
2 dff1o4 5375 . . . . . 6  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
32baib 904 . . . . 5  |-  ( F  Fn  A  ->  ( F : A -1-1-onto-> B  <->  `' F  Fn  B
) )
41, 3syl 14 . . . 4  |-  ( F : A --> B  -> 
( F : A -1-1-onto-> B  <->  `' F  Fn  B ) )
5 fnres 5239 . . . . . 6  |-  ( ( `' F  |`  B )  Fn  B  <->  A. y  e.  B  E! z 
y `' F z )
6 nfcv 2281 . . . . . . . . . 10  |-  F/_ x
z
7 fmpt.1 . . . . . . . . . . 11  |-  F  =  ( x  e.  A  |->  C )
8 nfmpt1 4021 . . . . . . . . . . 11  |-  F/_ x
( x  e.  A  |->  C )
97, 8nfcxfr 2278 . . . . . . . . . 10  |-  F/_ x F
10 nfcv 2281 . . . . . . . . . 10  |-  F/_ x
y
116, 9, 10nfbr 3974 . . . . . . . . 9  |-  F/ x  z F y
12 nfv 1508 . . . . . . . . 9  |-  F/ z ( x  e.  A  /\  y  =  C
)
13 breq1 3932 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z F y  <->  x F
y ) )
14 df-mpt 3991 . . . . . . . . . . . . 13  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
157, 14eqtri 2160 . . . . . . . . . . . 12  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
1615breqi 3935 . . . . . . . . . . 11  |-  ( x F y  <->  x { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) } y )
17 df-br 3930 . . . . . . . . . . . 12  |-  ( x { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } )
18 opabid 4179 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }  <->  ( x  e.  A  /\  y  =  C ) )
1917, 18bitri 183 . . . . . . . . . . 11  |-  ( x { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } y  <-> 
( x  e.  A  /\  y  =  C
) )
2016, 19bitri 183 . . . . . . . . . 10  |-  ( x F y  <->  ( x  e.  A  /\  y  =  C ) )
2113, 20syl6bb 195 . . . . . . . . 9  |-  ( z  =  x  ->  (
z F y  <->  ( x  e.  A  /\  y  =  C ) ) )
2211, 12, 21cbveu 2023 . . . . . . . 8  |-  ( E! z  z F y  <-> 
E! x ( x  e.  A  /\  y  =  C ) )
23 vex 2689 . . . . . . . . . 10  |-  y  e. 
_V
24 vex 2689 . . . . . . . . . 10  |-  z  e. 
_V
2523, 24brcnv 4722 . . . . . . . . 9  |-  ( y `' F z  <->  z F
y )
2625eubii 2008 . . . . . . . 8  |-  ( E! z  y `' F
z  <->  E! z  z F y )
27 df-reu 2423 . . . . . . . 8  |-  ( E! x  e.  A  y  =  C  <->  E! x
( x  e.  A  /\  y  =  C
) )
2822, 26, 273bitr4i 211 . . . . . . 7  |-  ( E! z  y `' F
z  <->  E! x  e.  A  y  =  C )
2928ralbii 2441 . . . . . 6  |-  ( A. y  e.  B  E! z  y `' F
z  <->  A. y  e.  B  E! x  e.  A  y  =  C )
305, 29bitri 183 . . . . 5  |-  ( ( `' F  |`  B )  Fn  B  <->  A. y  e.  B  E! x  e.  A  y  =  C )
31 relcnv 4917 . . . . . . 7  |-  Rel  `' F
32 df-rn 4550 . . . . . . . 8  |-  ran  F  =  dom  `' F
33 frn 5281 . . . . . . . 8  |-  ( F : A --> B  ->  ran  F  C_  B )
3432, 33eqsstrrid 3144 . . . . . . 7  |-  ( F : A --> B  ->  dom  `' F  C_  B )
35 relssres 4857 . . . . . . 7  |-  ( ( Rel  `' F  /\  dom  `' F  C_  B )  ->  ( `' F  |`  B )  =  `' F )
3631, 34, 35sylancr 410 . . . . . 6  |-  ( F : A --> B  -> 
( `' F  |`  B )  =  `' F )
3736fneq1d 5213 . . . . 5  |-  ( F : A --> B  -> 
( ( `' F  |`  B )  Fn  B  <->  `' F  Fn  B ) )
3830, 37syl5bbr 193 . . . 4  |-  ( F : A --> B  -> 
( A. y  e.  B  E! x  e.  A  y  =  C  <->  `' F  Fn  B
) )
394, 38bitr4d 190 . . 3  |-  ( F : A --> B  -> 
( F : A -1-1-onto-> B  <->  A. y  e.  B  E! x  e.  A  y  =  C ) )
4039pm5.32i 449 . 2  |-  ( ( F : A --> B  /\  F : A -1-1-onto-> B )  <->  ( F : A --> B  /\  A. y  e.  B  E! x  e.  A  y  =  C ) )
41 f1of 5367 . . 3  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
4241pm4.71ri 389 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A
--> B  /\  F : A
-1-1-onto-> B ) )
437fmpt 5570 . . 3  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
4443anbi1i 453 . 2  |-  ( ( A. x  e.  A  C  e.  B  /\  A. y  e.  B  E! x  e.  A  y  =  C )  <->  ( F : A --> B  /\  A. y  e.  B  E! x  e.  A  y  =  C ) )
4540, 42, 443bitr4i 211 1  |-  ( F : A -1-1-onto-> B  <->  ( A. x  e.  A  C  e.  B  /\  A. y  e.  B  E! x  e.  A  y  =  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E!weu 1999   A.wral 2416   E!wreu 2418    C_ wss 3071   <.cop 3530   class class class wbr 3929   {copab 3988    |-> cmpt 3989   `'ccnv 4538   dom cdm 4539   ran crn 4540    |` cres 4541   Rel wrel 4544    Fn wfn 5118   -->wf 5119   -1-1-onto->wf1o 5122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131
This theorem is referenced by:  xpf1o  6738  icoshftf1o  9781
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