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Theorem f1o3d 27144
Description: Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)
Hypotheses
Ref Expression
f1o3d.1  |-  ( ph  ->  F  =  ( x  e.  A  |->  C ) )
f1o3d.2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
f1o3d.3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
f1o3d.4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
Assertion
Ref Expression
f1o3d  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
Distinct variable groups:    x, y, A    x, B, y    y, C    x, D    ph, x, y
Allowed substitution hints:    C( x)    D( y)    F( x, y)

Proof of Theorem f1o3d
StepHypRef Expression
1 f1o3d.2 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
21ralrimiva 2878 . . . . 5  |-  ( ph  ->  A. x  e.  A  C  e.  B )
3 eqid 2467 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
43fnmpt 5705 . . . . 5  |-  ( A. x  e.  A  C  e.  B  ->  ( x  e.  A  |->  C )  Fn  A )
52, 4syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  C )  Fn  A
)
6 f1o3d.1 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  C ) )
76fneq1d 5669 . . . 4  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  C )  Fn  A ) )
85, 7mpbird 232 . . 3  |-  ( ph  ->  F  Fn  A )
9 f1o3d.3 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
109ralrimiva 2878 . . . . 5  |-  ( ph  ->  A. y  e.  B  D  e.  A )
11 eqid 2467 . . . . . 6  |-  ( y  e.  B  |->  D )  =  ( y  e.  B  |->  D )
1211fnmpt 5705 . . . . 5  |-  ( A. y  e.  B  D  e.  A  ->  ( y  e.  B  |->  D )  Fn  B )
1310, 12syl 16 . . . 4  |-  ( ph  ->  ( y  e.  B  |->  D )  Fn  B
)
14 eleq1a 2550 . . . . . . . . . . 11  |-  ( C  e.  B  ->  (
y  =  C  -> 
y  e.  B ) )
151, 14syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  C  -> 
y  e.  B ) )
1615impr 619 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
y  e.  B )
17 f1o3d.4 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
1817biimpar 485 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  y  =  C )  ->  x  =  D )
1918exp42 611 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( y  =  C  ->  x  =  D ) ) ) )
2019com34 83 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  A  ->  ( y  =  C  ->  ( y  e.  B  ->  x  =  D ) ) ) )
2120imp32 433 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  ->  x  =  D ) )
2216, 21jcai 536 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  /\  x  =  D
) )
23 eleq1a 2550 . . . . . . . . . . 11  |-  ( D  e.  A  ->  (
x  =  D  ->  x  e.  A )
)
249, 23syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  B )  ->  (
x  =  D  ->  x  e.  A )
)
2524impr 619 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  ->  x  e.  A )
2617biimpa 484 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  x  =  D )  ->  y  =  C )
2726exp42 611 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( x  =  D  ->  y  =  C ) ) ) )
2827com23 78 . . . . . . . . . . 11  |-  ( ph  ->  ( y  e.  B  ->  ( x  e.  A  ->  ( x  =  D  ->  y  =  C ) ) ) )
2928com34 83 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  B  ->  ( x  =  D  ->  ( x  e.  A  ->  y  =  C ) ) ) )
3029imp32 433 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  -> 
( x  e.  A  ->  y  =  C ) )
3125, 30jcai 536 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  -> 
( x  e.  A  /\  y  =  C
) )
3222, 31impbida 830 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
3332opabbidv 4510 . . . . . 6  |-  ( ph  ->  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( y  e.  B  /\  x  =  D ) } )
34 df-mpt 4507 . . . . . . . . 9  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
356, 34syl6eq 2524 . . . . . . . 8  |-  ( ph  ->  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) } )
3635cnveqd 5176 . . . . . . 7  |-  ( ph  ->  `' F  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) } )
37 cnvopab 5405 . . . . . . 7  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }
3836, 37syl6eq 2524 . . . . . 6  |-  ( ph  ->  `' F  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C
) } )
39 df-mpt 4507 . . . . . . 7  |-  ( y  e.  B  |->  D )  =  { <. y ,  x >.  |  (
y  e.  B  /\  x  =  D ) }
4039a1i 11 . . . . . 6  |-  ( ph  ->  ( y  e.  B  |->  D )  =  { <. y ,  x >.  |  ( y  e.  B  /\  x  =  D
) } )
4133, 38, 403eqtr4d 2518 . . . . 5  |-  ( ph  ->  `' F  =  (
y  e.  B  |->  D ) )
4241fneq1d 5669 . . . 4  |-  ( ph  ->  ( `' F  Fn  B 
<->  ( y  e.  B  |->  D )  Fn  B
) )
4313, 42mpbird 232 . . 3  |-  ( ph  ->  `' F  Fn  B
)
44 dff1o4 5822 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
458, 43, 44sylanbrc 664 . 2  |-  ( ph  ->  F : A -1-1-onto-> B )
4645, 41jca 532 1  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {copab 4504    |-> cmpt 4505   `'ccnv 4998    Fn wfn 5581   -1-1-onto->wf1o 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593
This theorem is referenced by:  ballotlemsf1o  28092
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