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Theorem supiso 7925
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 6219 . . . 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 210 . 2  |-  ( ph  ->  S  Or  B )
6 isof1o 6196 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  F : A -1-1-onto-> B
)
7 f1of 5798 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
82, 6, 73syl 20 . . 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 7909 . . 3  |-  ( ph  ->  sup ( C ,  A ,  R )  e.  A )
118, 10ffvelrnd 6008 . 2  |-  ( ph  ->  ( F `  sup ( C ,  A ,  R ) )  e.  B )
121, 9supub 7910 . . . . . 6  |-  ( ph  ->  ( u  e.  C  ->  -.  sup ( C ,  A ,  R
) R u ) )
1312ralrimiv 2866 . . . . 5  |-  ( ph  ->  A. u  e.  C  -.  sup ( C ,  A ,  R ) R u )
141, 9suplub 7911 . . . . . . 7  |-  ( ph  ->  ( ( u  e.  A  /\  u R sup ( C ,  A ,  R )
)  ->  E. z  e.  C  u R
z ) )
1514expd 434 . . . . . 6  |-  ( ph  ->  ( u  e.  A  ->  ( u R sup ( C ,  A ,  R )  ->  E. z  e.  C  u R
z ) ) )
1615ralrimiv 2866 . . . . 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 7923 . . . . . 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 666 . . . . 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 919 . . . 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 457 . . 3  |-  ( ph  ->  A. w  e.  ( F " C )  -.  ( F `  sup ( C ,  A ,  R ) ) S w )
2221r19.21bi 2823 . 2  |-  ( (
ph  /\  w  e.  ( F " C ) )  ->  -.  ( F `  sup ( C ,  A ,  R
) ) S w )
2320simprd 461 . . . 4  |-  ( ph  ->  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. v  e.  ( F " C ) w S v ) )
2423r19.21bi 2823 . . 3  |-  ( (
ph  /\  w  e.  B )  ->  (
w S ( F `
 sup ( C ,  A ,  R
) )  ->  E. v  e.  ( F " C
) w S v ) )
2524impr 617 . 2  |-  ( (
ph  /\  ( w  e.  B  /\  w S ( F `  sup ( C ,  A ,  R ) ) ) )  ->  E. v  e.  ( F " C
) w S v )
265, 11, 22, 25eqsupd 7908 1  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461   class class class wbr 4439    Or wor 4788   "cima 4991   -->wf 5566   -1-1-onto->wf1o 5569   ` cfv 5570    Isom wiso 5571   supcsup 7892
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-8 1825  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-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  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-reu 2811  df-rmo 2812  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-po 4789  df-so 4790  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  df-isom 5579  df-riota 6232  df-sup 7893
This theorem is referenced by:  infmsup  10516
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