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Theorem supiso 7478
Description: Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypotheses
Ref Expression
supiso.1  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
supiso.2  |-  ( ph  ->  C  C_  A )
supisoex.3  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )
supiso.4  |-  ( ph  ->  R  Or  A )
Assertion
Ref Expression
supiso  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
Distinct variable groups:    x, y,
z, A    x, C, y, z    x, F, y, z    x, R, y, z    x, S, y, z    x, B, y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem supiso
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supiso.4 . . 3  |-  ( ph  ->  R  Or  A )
2 supiso.1 . . . 4  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
3 isoso 6069 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
<->  S  Or  B ) )
42, 3syl 16 . . 3  |-  ( ph  ->  ( R  Or  A  <->  S  Or  B ) )
51, 4mpbid 203 . 2  |-  ( ph  ->  S  Or  B )
6 isof1o 6046 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  F : A -1-1-onto-> B
)
7 f1of 5675 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
82, 6, 73syl 19 . . 3  |-  ( ph  ->  F : A --> B )
9 supisoex.3 . . . 4  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )
101, 9supcl 7464 . . 3  |-  ( ph  ->  sup ( C ,  A ,  R )  e.  A )
118, 10ffvelrnd 5872 . 2  |-  ( ph  ->  ( F `  sup ( C ,  A ,  R ) )  e.  B )
121, 9supub 7465 . . . . . 6  |-  ( ph  ->  ( u  e.  C  ->  -.  sup ( C ,  A ,  R
) R u ) )
1312ralrimiv 2789 . . . . 5  |-  ( ph  ->  A. u  e.  C  -.  sup ( C ,  A ,  R ) R u )
141, 9suplub 7466 . . . . . . 7  |-  ( ph  ->  ( ( u  e.  A  /\  u R sup ( C ,  A ,  R )
)  ->  E. z  e.  C  u R
z ) )
1514exp3a 427 . . . . . 6  |-  ( ph  ->  ( u  e.  A  ->  ( u R sup ( C ,  A ,  R )  ->  E. z  e.  C  u R
z ) ) )
1615ralrimiv 2789 . . . . 5  |-  ( ph  ->  A. u  e.  A  ( u R sup ( C ,  A ,  R )  ->  E. z  e.  C  u R
z ) )
17 supiso.2 . . . . . . 7  |-  ( ph  ->  C  C_  A )
182, 17supisolem 7476 . . . . . 6  |-  ( (
ph  /\  sup ( C ,  A ,  R )  e.  A
)  ->  ( ( A. u  e.  C  -.  sup ( C ,  A ,  R ) R u  /\  A. u  e.  A  ( u R sup ( C ,  A ,  R )  ->  E. z  e.  C  u R z ) )  <-> 
( A. w  e.  ( F " C
)  -.  ( F `
 sup ( C ,  A ,  R
) ) S w  /\  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. v  e.  ( F " C ) w S v ) ) ) )
1910, 18mpdan 651 . . . . 5  |-  ( ph  ->  ( ( A. u  e.  C  -.  sup ( C ,  A ,  R ) R u  /\  A. u  e.  A  ( u R sup ( C ,  A ,  R )  ->  E. z  e.  C  u R z ) )  <-> 
( A. w  e.  ( F " C
)  -.  ( F `
 sup ( C ,  A ,  R
) ) S w  /\  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. v  e.  ( F " C ) w S v ) ) ) )
2013, 16, 19mpbi2and 889 . . . 4  |-  ( ph  ->  ( A. w  e.  ( F " C
)  -.  ( F `
 sup ( C ,  A ,  R
) ) S w  /\  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. v  e.  ( F " C ) w S v ) ) )
2120simpld 447 . . 3  |-  ( ph  ->  A. w  e.  ( F " C )  -.  ( F `  sup ( C ,  A ,  R ) ) S w )
2221r19.21bi 2805 . 2  |-  ( (
ph  /\  w  e.  ( F " C ) )  ->  -.  ( F `  sup ( C ,  A ,  R
) ) S w )
2320simprd 451 . . . 4  |-  ( ph  ->  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. v  e.  ( F " C ) w S v ) )
2423r19.21bi 2805 . . 3  |-  ( (
ph  /\  w  e.  B )  ->  (
w S ( F `
 sup ( C ,  A ,  R
) )  ->  E. v  e.  ( F " C
) w S v ) )
2524impr 604 . 2  |-  ( (
ph  /\  ( w  e.  B  /\  w S ( F `  sup ( C ,  A ,  R ) ) ) )  ->  E. v  e.  ( F " C
) w S v )
265, 11, 22, 25eqsupd 7463 1  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707    C_ wss 3321   class class class wbr 4213    Or wor 4503   "cima 4882   -->wf 5451   -1-1-onto->wf1o 5454   ` cfv 5455    Isom wiso 5456   supcsup 7446
This theorem is referenced by:  infmsup  9987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-riota 6550  df-sup 7447
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