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Theorem f1oresrab 6044
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 5809 . . . 4  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
3 funcnvcnv 5637 . . . 4  |-  ( Fun 
F  ->  Fun  `' `' F )
41, 2, 33syl 20 . . 3  |-  ( ph  ->  Fun  `' `' F
)
5 f1ocnv 5819 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
61, 5syl 16 . . . . . 6  |-  ( ph  ->  `' F : B -1-1-onto-> A )
7 f1of1 5806 . . . . . 6  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B -1-1-> A
)
86, 7syl 16 . . . . 5  |-  ( ph  ->  `' F : B -1-1-> A
)
9 ssrab2 3578 . . . . 5  |-  { y  e.  B  |  ch }  C_  B
10 f1ores 5821 . . . . 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 } ) )
118, 9, 10sylancl 662 . . . 4  |-  ( ph  ->  ( `' F  |`  { y  e.  B  |  ch } ) : { y  e.  B  |  ch } -1-1-onto-> ( `' F " { y  e.  B  |  ch } ) )
12 f1oresrab.1 . . . . . . 7  |-  F  =  ( x  e.  A  |->  C )
1312mptpreima 5491 . . . . . 6  |-  ( `' F " { y  e.  B  |  ch } )  =  {
x  e.  A  |  C  e.  { y  e.  B  |  ch } }
14 f1oresrab.3 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A  /\  y  =  C )  ->  ( ch  <->  ps ) )
15143expia 1193 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  C  -> 
( ch  <->  ps )
) )
1615alrimiv 1690 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  =  C  ->  ( ch  <->  ps )
) )
17 f1of 5807 . . . . . . . . . . 11  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
181, 17syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
1912fmpt 6033 . . . . . . . . . 10  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
2018, 19sylibr 212 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  C  e.  B )
2120r19.21bi 2826 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
22 elrab3t 3253 . . . . . . . 8  |-  ( ( A. y ( y  =  C  ->  ( ch 
<->  ps ) )  /\  C  e.  B )  ->  ( C  e.  {
y  e.  B  |  ch }  <->  ps ) )
2316, 21, 22syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( C  e.  { y  e.  B  |  ch } 
<->  ps ) )
2423rabbidva 3097 . . . . . 6  |-  ( ph  ->  { x  e.  A  |  C  e.  { y  e.  B  |  ch } }  =  {
x  e.  A  |  ps } )
2513, 24syl5eq 2513 . . . . 5  |-  ( ph  ->  ( `' F " { y  e.  B  |  ch } )  =  { x  e.  A  |  ps } )
26 f1oeq3 5800 . . . . 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 } ) )
2725, 26syl 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 } ) )
2811, 27mpbid 210 . . 3  |-  ( ph  ->  ( `' F  |`  { y  e.  B  |  ch } ) : { y  e.  B  |  ch } -1-1-onto-> { x  e.  A  |  ps } )
29 f1orescnv 5822 . . 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 } )
304, 28, 29syl2anc 661 . 2  |-  ( ph  ->  ( `' `' F  |` 
{ x  e.  A  |  ps } ) : { x  e.  A  |  ps } -1-1-onto-> { y  e.  B  |  ch } )
31 rescnvcnv 5461 . . 3  |-  ( `' `' F  |`  { x  e.  A  |  ps } )  =  ( F  |`  { x  e.  A  |  ps } )
32 f1oeq1 5798 . . 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 } ) )
3331, 32ax-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 } )
3430, 33sylib 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 369    /\ w3a 968   A.wal 1372    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811    C_ wss 3469    |-> cmpt 4498   `'ccnv 4991    |` cres 4994   "cima 4995   Fun wfun 5573   -->wf 5575   -1-1->wf1 5576   -1-1-onto->wf1o 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587
This theorem is referenced by:  wlknwwlknvbij  24402  clwwlkvbij  24463  fpwrelmapffs  27215  eulerpartlemn  27946
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