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Theorem f1oresrab 6039
Description: Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
Hypotheses
Ref Expression
f1oresrab.1  |-  F  =  ( x  e.  A  |->  C )
f1oresrab.2  |-  ( ph  ->  F : A -1-1-onto-> B )
f1oresrab.3  |-  ( (
ph  /\  x  e.  A  /\  y  =  C )  ->  ( ch  <->  ps ) )
Assertion
Ref Expression
f1oresrab  |-  ( ph  ->  ( F  |`  { x  e.  A  |  ps } ) : {
x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch } )
Distinct variable groups:    x, y, A    x, B, y    y, C    ph, x, y    ps, y    ch, x
Allowed substitution hints:    ps( x)    ch( y)    C( x)    F( x, y)

Proof of Theorem f1oresrab
StepHypRef Expression
1 f1oresrab.2 . . . 4  |-  ( ph  ->  F : A -1-1-onto-> B )
2 f1ofun 5800 . . . 4  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
3 funcnvcnv 5628 . . . 4  |-  ( Fun 
F  ->  Fun  `' `' F )
41, 2, 33syl 20 . . 3  |-  ( ph  ->  Fun  `' `' F
)
5 f1ocnv 5810 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
6 f1of1 5797 . . . . . 6  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B -1-1-> A
)
71, 5, 63syl 20 . . . . 5  |-  ( ph  ->  `' F : B -1-1-> A
)
8 ssrab2 3571 . . . . 5  |-  { y  e.  B  |  ch }  C_  B
9 f1ores 5812 . . . . 5  |-  ( ( `' F : B -1-1-> A  /\  { y  e.  B  |  ch }  C_  B
)  ->  ( `' F  |`  { y  e.  B  |  ch }
) : { y  e.  B  |  ch }
-1-1-onto-> ( `' F " { y  e.  B  |  ch } ) )
107, 8, 9sylancl 660 . . . 4  |-  ( ph  ->  ( `' F  |`  { y  e.  B  |  ch } ) : { y  e.  B  |  ch } -1-1-onto-> ( `' F " { y  e.  B  |  ch } ) )
11 f1oresrab.1 . . . . . . 7  |-  F  =  ( x  e.  A  |->  C )
1211mptpreima 5483 . . . . . 6  |-  ( `' F " { y  e.  B  |  ch } )  =  {
x  e.  A  |  C  e.  { y  e.  B  |  ch } }
13 f1oresrab.3 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A  /\  y  =  C )  ->  ( ch  <->  ps ) )
14133expia 1196 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  C  -> 
( ch  <->  ps )
) )
1514alrimiv 1724 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  =  C  ->  ( ch  <->  ps )
) )
16 f1of 5798 . . . . . . . . . . 11  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
171, 16syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
1811fmpt 6028 . . . . . . . . . 10  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
1917, 18sylibr 212 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  C  e.  B )
2019r19.21bi 2823 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
21 elrab3t 3253 . . . . . . . 8  |-  ( ( A. y ( y  =  C  ->  ( ch 
<->  ps ) )  /\  C  e.  B )  ->  ( C  e.  {
y  e.  B  |  ch }  <->  ps ) )
2215, 20, 21syl2anc 659 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( C  e.  { y  e.  B  |  ch } 
<->  ps ) )
2322rabbidva 3097 . . . . . 6  |-  ( ph  ->  { x  e.  A  |  C  e.  { y  e.  B  |  ch } }  =  {
x  e.  A  |  ps } )
2412, 23syl5eq 2507 . . . . 5  |-  ( ph  ->  ( `' F " { y  e.  B  |  ch } )  =  { x  e.  A  |  ps } )
25 f1oeq3 5791 . . . . 5  |-  ( ( `' F " { y  e.  B  |  ch } )  =  {
x  e.  A  |  ps }  ->  ( ( `' F  |`  { y  e.  B  |  ch } ) : {
y  e.  B  |  ch } -1-1-onto-> ( `' F " { y  e.  B  |  ch } )  <->  ( `' F  |`  { y  e.  B  |  ch }
) : { y  e.  B  |  ch }
-1-1-onto-> { x  e.  A  |  ps } ) )
2624, 25syl 16 . . . 4  |-  ( ph  ->  ( ( `' F  |` 
{ y  e.  B  |  ch } ) : { y  e.  B  |  ch } -1-1-onto-> ( `' F " { y  e.  B  |  ch } )  <->  ( `' F  |`  { y  e.  B  |  ch }
) : { y  e.  B  |  ch }
-1-1-onto-> { x  e.  A  |  ps } ) )
2710, 26mpbid 210 . . 3  |-  ( ph  ->  ( `' F  |`  { y  e.  B  |  ch } ) : { y  e.  B  |  ch } -1-1-onto-> { x  e.  A  |  ps } )
28 f1orescnv 5813 . . 3  |-  ( ( Fun  `' `' F  /\  ( `' F  |`  { y  e.  B  |  ch } ) : { y  e.  B  |  ch } -1-1-onto-> { x  e.  A  |  ps } )  -> 
( `' `' F  |` 
{ x  e.  A  |  ps } ) : { x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch } )
294, 27, 28syl2anc 659 . 2  |-  ( ph  ->  ( `' `' F  |` 
{ x  e.  A  |  ps } ) : { x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch } )
30 rescnvcnv 5453 . . 3  |-  ( `' `' F  |`  { x  e.  A  |  ps } )  =  ( F  |`  { x  e.  A  |  ps } )
31 f1oeq1 5789 . . 3  |-  ( ( `' `' F  |`  { x  e.  A  |  ps } )  =  ( F  |`  { x  e.  A  |  ps } )  ->  (
( `' `' F  |` 
{ x  e.  A  |  ps } ) : { x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch }  <->  ( F  |` 
{ x  e.  A  |  ps } ) : { x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch } ) )
3230, 31ax-mp 5 . 2  |-  ( ( `' `' F  |`  { x  e.  A  |  ps } ) : {
x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch }  <->  ( F  |` 
{ x  e.  A  |  ps } ) : { x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch } )
3329, 32sylib 196 1  |-  ( ph  ->  ( F  |`  { x  e.  A  |  ps } ) : {
x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   A.wal 1396    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808    C_ wss 3461    |-> cmpt 4497   `'ccnv 4987    |` cres 4990   "cima 4991   Fun wfun 5564   -->wf 5566   -1-1->wf1 5567   -1-1-onto->wf1o 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578
This theorem is referenced by:  wlknwwlknvbij  24945  clwwlkvbij  25006  rabfodom  27606  fpwrelmapffs  27791  eulerpartlemn  28587
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