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Theorem f1ompt 5348
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 5074 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
2 dff1o4 5162 . . . . . 6  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
32baib 839 . . . . 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 5043 . . . . . 6  |-  ( ( `' F  |`  B )  Fn  B  <->  A. y  e.  B  E! z 
y `' F z )
6 nfcv 2194 . . . . . . . . . 10  |-  F/_ x
z
7 fmpt.1 . . . . . . . . . . 11  |-  F  =  ( x  e.  A  |->  C )
8 nfmpt1 3878 . . . . . . . . . . 11  |-  F/_ x
( x  e.  A  |->  C )
97, 8nfcxfr 2191 . . . . . . . . . 10  |-  F/_ x F
10 nfcv 2194 . . . . . . . . . 10  |-  F/_ x
y
116, 9, 10nfbr 3836 . . . . . . . . 9  |-  F/ x  z F y
12 nfv 1437 . . . . . . . . 9  |-  F/ z ( x  e.  A  /\  y  =  C
)
13 breq1 3795 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z F y  <->  x F
y ) )
14 df-mpt 3848 . . . . . . . . . . . . 13  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
157, 14eqtri 2076 . . . . . . . . . . . 12  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
1615breqi 3798 . . . . . . . . . . 11  |-  ( x F y  <->  x { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) } y )
17 df-br 3793 . . . . . . . . . . . 12  |-  ( x { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } )
18 opabid 4022 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }  <->  ( x  e.  A  /\  y  =  C ) )
1917, 18bitri 177 . . . . . . . . . . 11  |-  ( x { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } y  <-> 
( x  e.  A  /\  y  =  C
) )
2016, 19bitri 177 . . . . . . . . . 10  |-  ( x F y  <->  ( x  e.  A  /\  y  =  C ) )
2113, 20syl6bb 189 . . . . . . . . 9  |-  ( z  =  x  ->  (
z F y  <->  ( x  e.  A  /\  y  =  C ) ) )
2211, 12, 21cbveu 1940 . . . . . . . 8  |-  ( E! z  z F y  <-> 
E! x ( x  e.  A  /\  y  =  C ) )
23 vex 2577 . . . . . . . . . 10  |-  y  e. 
_V
24 vex 2577 . . . . . . . . . 10  |-  z  e. 
_V
2523, 24brcnv 4546 . . . . . . . . 9  |-  ( y `' F z  <->  z F
y )
2625eubii 1925 . . . . . . . 8  |-  ( E! z  y `' F
z  <->  E! z  z F y )
27 df-reu 2330 . . . . . . . 8  |-  ( E! x  e.  A  y  =  C  <->  E! x
( x  e.  A  /\  y  =  C
) )
2822, 26, 273bitr4i 205 . . . . . . 7  |-  ( E! z  y `' F
z  <->  E! x  e.  A  y  =  C )
2928ralbii 2347 . . . . . 6  |-  ( A. y  e.  B  E! z  y `' F
z  <->  A. y  e.  B  E! x  e.  A  y  =  C )
305, 29bitri 177 . . . . 5  |-  ( ( `' F  |`  B )  Fn  B  <->  A. y  e.  B  E! x  e.  A  y  =  C )
31 relcnv 4731 . . . . . . 7  |-  Rel  `' F
32 df-rn 4384 . . . . . . . 8  |-  ran  F  =  dom  `' F
33 frn 5080 . . . . . . . 8  |-  ( F : A --> B  ->  ran  F  C_  B )
3432, 33syl5eqssr 3018 . . . . . . 7  |-  ( F : A --> B  ->  dom  `' F  C_  B )
35 relssres 4676 . . . . . . 7  |-  ( ( Rel  `' F  /\  dom  `' F  C_  B )  ->  ( `' F  |`  B )  =  `' F )
3631, 34, 35sylancr 399 . . . . . 6  |-  ( F : A --> B  -> 
( `' F  |`  B )  =  `' F )
3736fneq1d 5017 . . . . 5  |-  ( F : A --> B  -> 
( ( `' F  |`  B )  Fn  B  <->  `' F  Fn  B ) )
3830, 37syl5bbr 187 . . . 4  |-  ( F : A --> B  -> 
( A. y  e.  B  E! x  e.  A  y  =  C  <->  `' F  Fn  B
) )
394, 38bitr4d 184 . . 3  |-  ( F : A --> B  -> 
( F : A -1-1-onto-> B  <->  A. y  e.  B  E! x  e.  A  y  =  C ) )
4039pm5.32i 435 . 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 5154 . . 3  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
4241pm4.71ri 378 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A
--> B  /\  F : A
-1-1-onto-> B ) )
437fmpt 5347 . . 3  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
4443anbi1i 439 . 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 205 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 101    <-> wb 102    = wceq 1259    e. wcel 1409   E!weu 1916   A.wral 2323   E!wreu 2325    C_ wss 2945   <.cop 3406   class class class wbr 3792   {copab 3845    |-> cmpt 3846   `'ccnv 4372   dom cdm 4373   ran crn 4374    |` cres 4375   Rel wrel 4378    Fn wfn 4925   -->wf 4926   -1-1-onto->wf1o 4929
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-rab 2332  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  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-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938
This theorem is referenced by:  icoshftf1o  8960
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