MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1ompt Structured version   Unicode version

Theorem f1ompt 5865
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 5559 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
2 dff1o4 5649 . . . . . 6  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
32baib 896 . . . . 5  |-  ( F  Fn  A  ->  ( F : A -1-1-onto-> B  <->  `' F  Fn  B
) )
41, 3syl 16 . . . 4  |-  ( F : A --> B  -> 
( F : A -1-1-onto-> B  <->  `' F  Fn  B ) )
5 fnres 5527 . . . . . 6  |-  ( ( `' F  |`  B )  Fn  B  <->  A. y  e.  B  E! z 
y `' F z )
6 nfcv 2579 . . . . . . . . . 10  |-  F/_ x
z
7 fmpt.1 . . . . . . . . . . 11  |-  F  =  ( x  e.  A  |->  C )
8 nfmpt1 4381 . . . . . . . . . . 11  |-  F/_ x
( x  e.  A  |->  C )
97, 8nfcxfr 2576 . . . . . . . . . 10  |-  F/_ x F
10 nfcv 2579 . . . . . . . . . 10  |-  F/_ x
y
116, 9, 10nfbr 4336 . . . . . . . . 9  |-  F/ x  z F y
12 nfv 1673 . . . . . . . . 9  |-  F/ z ( x  e.  A  /\  y  =  C
)
13 breq1 4295 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z F y  <->  x F
y ) )
14 df-mpt 4352 . . . . . . . . . . . . 13  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
157, 14eqtri 2463 . . . . . . . . . . . 12  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
1615breqi 4298 . . . . . . . . . . 11  |-  ( x F y  <->  x { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) } y )
17 df-br 4293 . . . . . . . . . . . 12  |-  ( x { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } )
18 opabid 4596 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }  <->  ( x  e.  A  /\  y  =  C ) )
1917, 18bitri 249 . . . . . . . . . . 11  |-  ( x { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } y  <-> 
( x  e.  A  /\  y  =  C
) )
2016, 19bitri 249 . . . . . . . . . 10  |-  ( x F y  <->  ( x  e.  A  /\  y  =  C ) )
2113, 20syl6bb 261 . . . . . . . . 9  |-  ( z  =  x  ->  (
z F y  <->  ( x  e.  A  /\  y  =  C ) ) )
2211, 12, 21cbveu 2296 . . . . . . . 8  |-  ( E! z  z F y  <-> 
E! x ( x  e.  A  /\  y  =  C ) )
23 vex 2975 . . . . . . . . . 10  |-  y  e. 
_V
24 vex 2975 . . . . . . . . . 10  |-  z  e. 
_V
2523, 24brcnv 5022 . . . . . . . . 9  |-  ( y `' F z  <->  z F
y )
2625eubii 2278 . . . . . . . 8  |-  ( E! z  y `' F
z  <->  E! z  z F y )
27 df-reu 2722 . . . . . . . 8  |-  ( E! x  e.  A  y  =  C  <->  E! x
( x  e.  A  /\  y  =  C
) )
2822, 26, 273bitr4i 277 . . . . . . 7  |-  ( E! z  y `' F
z  <->  E! x  e.  A  y  =  C )
2928ralbii 2739 . . . . . 6  |-  ( A. y  e.  B  E! z  y `' F
z  <->  A. y  e.  B  E! x  e.  A  y  =  C )
305, 29bitri 249 . . . . 5  |-  ( ( `' F  |`  B )  Fn  B  <->  A. y  e.  B  E! x  e.  A  y  =  C )
31 relcnv 5206 . . . . . . 7  |-  Rel  `' F
32 df-rn 4851 . . . . . . . 8  |-  ran  F  =  dom  `' F
33 frn 5565 . . . . . . . 8  |-  ( F : A --> B  ->  ran  F  C_  B )
3432, 33syl5eqssr 3401 . . . . . . 7  |-  ( F : A --> B  ->  dom  `' F  C_  B )
35 relssres 5147 . . . . . . 7  |-  ( ( Rel  `' F  /\  dom  `' F  C_  B )  ->  ( `' F  |`  B )  =  `' F )
3631, 34, 35sylancr 663 . . . . . 6  |-  ( F : A --> B  -> 
( `' F  |`  B )  =  `' F )
3736fneq1d 5501 . . . . 5  |-  ( F : A --> B  -> 
( ( `' F  |`  B )  Fn  B  <->  `' F  Fn  B ) )
3830, 37syl5bbr 259 . . . 4  |-  ( F : A --> B  -> 
( A. y  e.  B  E! x  e.  A  y  =  C  <->  `' F  Fn  B
) )
394, 38bitr4d 256 . . 3  |-  ( F : A --> B  -> 
( F : A -1-1-onto-> B  <->  A. y  e.  B  E! x  e.  A  y  =  C ) )
4039pm5.32i 637 . 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 5641 . . 3  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
4241pm4.71ri 633 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A
--> B  /\  F : A
-1-1-onto-> B ) )
437fmpt 5864 . . 3  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
4443anbi1i 695 . 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 277 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E!weu 2253   A.wral 2715   E!wreu 2717    C_ wss 3328   <.cop 3883   class class class wbr 4292   {copab 4349    e. cmpt 4350   `'ccnv 4839   dom cdm 4840   ran crn 4841    |` cres 4842   Rel wrel 4845    Fn wfn 5413   -->wf 5414   -1-1-onto->wf1o 5417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426
This theorem is referenced by:  oaf1o  7002  xpf1o  7473  icoshftf1o  11408  dfod2  16065  cusgrafilem2  23388  gsummptf1o  26247  xrmulc1cn  26360  fprodser  27462  numclwlk2lem2f1o  30698
  Copyright terms: Public domain W3C validator