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Theorem f1oresrab 6071
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 5830 . . . 4  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
3 funcnvcnv 5651 . . . 4  |-  ( Fun 
F  ->  Fun  `' `' F )
41, 2, 33syl 18 . . 3  |-  ( ph  ->  Fun  `' `' F
)
5 f1ocnv 5840 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
6 f1of1 5827 . . . . . 6  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B -1-1-> A
)
71, 5, 63syl 18 . . . . 5  |-  ( ph  ->  `' F : B -1-1-> A
)
8 ssrab2 3500 . . . . 5  |-  { y  e.  B  |  ch }  C_  B
9 f1ores 5842 . . . . 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 675 . . . 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 5335 . . . . . 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 1233 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  C  -> 
( ch  <->  ps )
) )
1514alrimiv 1781 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  =  C  ->  ( ch  <->  ps )
) )
16 f1of 5828 . . . . . . . . . . 11  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
171, 16syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
1811fmpt 6058 . . . . . . . . . 10  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
1917, 18sylibr 217 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  C  e.  B )
2019r19.21bi 2776 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
21 elrab3t 3183 . . . . . . . 8  |-  ( ( A. y ( y  =  C  ->  ( ch 
<->  ps ) )  /\  C  e.  B )  ->  ( C  e.  {
y  e.  B  |  ch }  <->  ps ) )
2215, 20, 21syl2anc 673 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( C  e.  { y  e.  B  |  ch } 
<->  ps ) )
2322rabbidva 3021 . . . . . 6  |-  ( ph  ->  { x  e.  A  |  C  e.  { y  e.  B  |  ch } }  =  {
x  e.  A  |  ps } )
2412, 23syl5eq 2517 . . . . 5  |-  ( ph  ->  ( `' F " { y  e.  B  |  ch } )  =  { x  e.  A  |  ps } )
25 f1oeq3 5820 . . . . 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 17 . . . 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 215 . . 3  |-  ( ph  ->  ( `' F  |`  { y  e.  B  |  ch } ) : { y  e.  B  |  ch } -1-1-onto-> { x  e.  A  |  ps } )
28 f1orescnv 5843 . . 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 673 . 2  |-  ( ph  ->  ( `' `' F  |` 
{ x  e.  A  |  ps } ) : { x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch } )
30 rescnvcnv 5305 . . 3  |-  ( `' `' F  |`  { x  e.  A  |  ps } )  =  ( F  |`  { x  e.  A  |  ps } )
31 f1oeq1 5818 . . 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 201 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 189    /\ wa 376    /\ w3a 1007   A.wal 1450    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760    C_ wss 3390    |-> cmpt 4454   `'ccnv 4838    |` cres 4841   "cima 4842   Fun wfun 5583   -->wf 5585   -1-1->wf1 5586   -1-1-onto->wf1o 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597
This theorem is referenced by:  wlknwwlknvbij  25547  clwwlkvbij  25608  rabfodom  28219  fpwrelmapffs  28394  eulerpartlemn  29287
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