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Theorem supisoti 6759
Description: Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
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 ) ) )
supisoti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
Assertion
Ref Expression
supisoti  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
Distinct variable groups:    v, u, x, y, z, A    u, C, v, x, y, z    ph, u    u, F, v, x, y, z    u, R, x, y, z    u, S, v, x, y, z   
u, B, v, x, y, z    v, R    ph, v, x
Allowed substitution hints:    ph( y, z)

Proof of Theorem supisoti
Dummy variables  w  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supisoti.ti . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
21ralrimivva 2456 . . . . . 6  |-  ( ph  ->  A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )
3 supiso.1 . . . . . . 7  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
4 isoti 6756 . . . . . . 7  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  <->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) ) )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  <->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) ) )
62, 5mpbid 146 . . . . 5  |-  ( ph  ->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) )
76r19.21bi 2462 . . . 4  |-  ( (
ph  /\  u  e.  B )  ->  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) )
87r19.21bi 2462 . . 3  |-  ( ( ( ph  /\  u  e.  B )  /\  v  e.  B )  ->  (
u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) )
98anasss 392 . 2  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u  =  v  <-> 
( -.  u S v  /\  -.  v S u ) ) )
10 isof1o 5600 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  F : A -1-1-onto-> B
)
11 f1of 5266 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
123, 10, 113syl 17 . . 3  |-  ( ph  ->  F : A --> B )
13 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 ) ) )
141, 13supclti 6747 . . 3  |-  ( ph  ->  sup ( C ,  A ,  R )  e.  A )
1512, 14ffvelrnd 5449 . 2  |-  ( ph  ->  ( F `  sup ( C ,  A ,  R ) )  e.  B )
161, 13supubti 6748 . . . . . 6  |-  ( ph  ->  ( j  e.  C  ->  -.  sup ( C ,  A ,  R
) R j ) )
1716ralrimiv 2446 . . . . 5  |-  ( ph  ->  A. j  e.  C  -.  sup ( C ,  A ,  R ) R j )
181, 13suplubti 6749 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  A  /\  j R sup ( C ,  A ,  R )
)  ->  E. z  e.  C  j R
z ) )
1918expd 255 . . . . . 6  |-  ( ph  ->  ( j  e.  A  ->  ( j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R
z ) ) )
2019ralrimiv 2446 . . . . 5  |-  ( ph  ->  A. j  e.  A  ( j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R
z ) )
21 supiso.2 . . . . . . 7  |-  ( ph  ->  C  C_  A )
223, 21supisolem 6757 . . . . . 6  |-  ( (
ph  /\  sup ( C ,  A ,  R )  e.  A
)  ->  ( ( A. j  e.  C  -.  sup ( C ,  A ,  R ) R j  /\  A. j  e.  A  (
j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j 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. k  e.  ( F " C ) w S k ) ) ) )
2314, 22mpdan 413 . . . . 5  |-  ( ph  ->  ( ( A. j  e.  C  -.  sup ( C ,  A ,  R ) R j  /\  A. j  e.  A  ( j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j 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. k  e.  ( F " C ) w S k ) ) ) )
2417, 20, 23mpbi2and 890 . . . 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. k  e.  ( F " C ) w S k ) ) )
2524simpld 111 . . 3  |-  ( ph  ->  A. w  e.  ( F " C )  -.  ( F `  sup ( C ,  A ,  R ) ) S w )
2625r19.21bi 2462 . 2  |-  ( (
ph  /\  w  e.  ( F " C ) )  ->  -.  ( F `  sup ( C ,  A ,  R
) ) S w )
2724simprd 113 . . . 4  |-  ( ph  ->  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. k  e.  ( F " C ) w S k ) )
2827r19.21bi 2462 . . 3  |-  ( (
ph  /\  w  e.  B )  ->  (
w S ( F `
 sup ( C ,  A ,  R
) )  ->  E. k  e.  ( F " C
) w S k ) )
2928impr 372 . 2  |-  ( (
ph  /\  ( w  e.  B  /\  w S ( F `  sup ( C ,  A ,  R ) ) ) )  ->  E. k  e.  ( F " C
) w S k )
309, 15, 26, 29eqsuptid 6746 1  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   A.wral 2360   E.wrex 2361    C_ wss 3000   class class class wbr 3851   "cima 4454   -->wf 5024   -1-1-onto->wf1o 5027   ` cfv 5028    Isom wiso 5029   supcsup 6731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-sup 6733
This theorem is referenced by:  infisoti  6781  infrenegsupex  9136
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