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Theorem f1ompt 5434
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 5147 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
2 dff1o4 5245 . . . . . 6  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
32baib 866 . . . . 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 5116 . . . . . 6  |-  ( ( `' F  |`  B )  Fn  B  <->  A. y  e.  B  E! z 
y `' F z )
6 nfcv 2228 . . . . . . . . . 10  |-  F/_ x
z
7 fmpt.1 . . . . . . . . . . 11  |-  F  =  ( x  e.  A  |->  C )
8 nfmpt1 3923 . . . . . . . . . . 11  |-  F/_ x
( x  e.  A  |->  C )
97, 8nfcxfr 2225 . . . . . . . . . 10  |-  F/_ x F
10 nfcv 2228 . . . . . . . . . 10  |-  F/_ x
y
116, 9, 10nfbr 3881 . . . . . . . . 9  |-  F/ x  z F y
12 nfv 1466 . . . . . . . . 9  |-  F/ z ( x  e.  A  /\  y  =  C
)
13 breq1 3840 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z F y  <->  x F
y ) )
14 df-mpt 3893 . . . . . . . . . . . . 13  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
157, 14eqtri 2108 . . . . . . . . . . . 12  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
1615breqi 3843 . . . . . . . . . . 11  |-  ( x F y  <->  x { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) } y )
17 df-br 3838 . . . . . . . . . . . 12  |-  ( x { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } )
18 opabid 4075 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }  <->  ( x  e.  A  /\  y  =  C ) )
1917, 18bitri 182 . . . . . . . . . . 11  |-  ( x { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } y  <-> 
( x  e.  A  /\  y  =  C
) )
2016, 19bitri 182 . . . . . . . . . 10  |-  ( x F y  <->  ( x  e.  A  /\  y  =  C ) )
2113, 20syl6bb 194 . . . . . . . . 9  |-  ( z  =  x  ->  (
z F y  <->  ( x  e.  A  /\  y  =  C ) ) )
2211, 12, 21cbveu 1972 . . . . . . . 8  |-  ( E! z  z F y  <-> 
E! x ( x  e.  A  /\  y  =  C ) )
23 vex 2622 . . . . . . . . . 10  |-  y  e. 
_V
24 vex 2622 . . . . . . . . . 10  |-  z  e. 
_V
2523, 24brcnv 4607 . . . . . . . . 9  |-  ( y `' F z  <->  z F
y )
2625eubii 1957 . . . . . . . 8  |-  ( E! z  y `' F
z  <->  E! z  z F y )
27 df-reu 2366 . . . . . . . 8  |-  ( E! x  e.  A  y  =  C  <->  E! x
( x  e.  A  /\  y  =  C
) )
2822, 26, 273bitr4i 210 . . . . . . 7  |-  ( E! z  y `' F
z  <->  E! x  e.  A  y  =  C )
2928ralbii 2384 . . . . . 6  |-  ( A. y  e.  B  E! z  y `' F
z  <->  A. y  e.  B  E! x  e.  A  y  =  C )
305, 29bitri 182 . . . . 5  |-  ( ( `' F  |`  B )  Fn  B  <->  A. y  e.  B  E! x  e.  A  y  =  C )
31 relcnv 4797 . . . . . . 7  |-  Rel  `' F
32 df-rn 4439 . . . . . . . 8  |-  ran  F  =  dom  `' F
33 frn 5155 . . . . . . . 8  |-  ( F : A --> B  ->  ran  F  C_  B )
3432, 33syl5eqssr 3069 . . . . . . 7  |-  ( F : A --> B  ->  dom  `' F  C_  B )
35 relssres 4737 . . . . . . 7  |-  ( ( Rel  `' F  /\  dom  `' F  C_  B )  ->  ( `' F  |`  B )  =  `' F )
3631, 34, 35sylancr 405 . . . . . 6  |-  ( F : A --> B  -> 
( `' F  |`  B )  =  `' F )
3736fneq1d 5090 . . . . 5  |-  ( F : A --> B  -> 
( ( `' F  |`  B )  Fn  B  <->  `' F  Fn  B ) )
3830, 37syl5bbr 192 . . . 4  |-  ( F : A --> B  -> 
( A. y  e.  B  E! x  e.  A  y  =  C  <->  `' F  Fn  B
) )
394, 38bitr4d 189 . . 3  |-  ( F : A --> B  -> 
( F : A -1-1-onto-> B  <->  A. y  e.  B  E! x  e.  A  y  =  C ) )
4039pm5.32i 442 . 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 5237 . . 3  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
4241pm4.71ri 384 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A
--> B  /\  F : A
-1-1-onto-> B ) )
437fmpt 5433 . . 3  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
4443anbi1i 446 . 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 210 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   E!weu 1948   A.wral 2359   E!wreu 2361    C_ wss 2997   <.cop 3444   class class class wbr 3837   {copab 3890    |-> cmpt 3891   `'ccnv 4427   dom cdm 4428   ran crn 4429    |` cres 4430   Rel wrel 4433    Fn wfn 4997   -->wf 4998   -1-1-onto->wf1o 5001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010
This theorem is referenced by:  xpf1o  6540  icoshftf1o  9377
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