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Theorem supiso 7223
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 5845 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
<->  S  Or  B ) )
42, 3syl 15 . . 3  |-  ( ph  ->  ( R  Or  A  <->  S  Or  B ) )
51, 4mpbid 201 . 2  |-  ( ph  ->  S  Or  B )
6 isof1o 5822 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  F : A -1-1-onto-> B
)
7 f1of 5472 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
82, 6, 73syl 18 . . 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 7209 . . 3  |-  ( ph  ->  sup ( C ,  A ,  R )  e.  A )
11 ffvelrn 5663 . . 3  |-  ( ( F : A --> B  /\  sup ( C ,  A ,  R )  e.  A
)  ->  ( F `  sup ( C ,  A ,  R )
)  e.  B )
128, 10, 11syl2anc 642 . 2  |-  ( ph  ->  ( F `  sup ( C ,  A ,  R ) )  e.  B )
131, 9supub 7210 . . . . . 6  |-  ( ph  ->  ( u  e.  C  ->  -.  sup ( C ,  A ,  R
) R u ) )
1413ralrimiv 2625 . . . . 5  |-  ( ph  ->  A. u  e.  C  -.  sup ( C ,  A ,  R ) R u )
151, 9suplub 7211 . . . . . . 7  |-  ( ph  ->  ( ( u  e.  A  /\  u R sup ( C ,  A ,  R )
)  ->  E. z  e.  C  u R
z ) )
1615exp3a 425 . . . . . 6  |-  ( ph  ->  ( u  e.  A  ->  ( u R sup ( C ,  A ,  R )  ->  E. z  e.  C  u R
z ) ) )
1716ralrimiv 2625 . . . . 5  |-  ( ph  ->  A. u  e.  A  ( u R sup ( C ,  A ,  R )  ->  E. z  e.  C  u R
z ) )
18 supiso.2 . . . . . . 7  |-  ( ph  ->  C  C_  A )
192, 18supisolem 7221 . . . . . 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 ) ) ) )
2010, 19mpdan 649 . . . . 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 ) ) ) )
2114, 17, 20mpbi2and 887 . . . 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 ) ) )
2221simpld 445 . . 3  |-  ( ph  ->  A. w  e.  ( F " C )  -.  ( F `  sup ( C ,  A ,  R ) ) S w )
2322r19.21bi 2641 . 2  |-  ( (
ph  /\  w  e.  ( F " C ) )  ->  -.  ( F `  sup ( C ,  A ,  R
) ) S w )
2421simprd 449 . . . 4  |-  ( ph  ->  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. v  e.  ( F " C ) w S v ) )
2524r19.21bi 2641 . . 3  |-  ( (
ph  /\  w  e.  B )  ->  (
w S ( F `
 sup ( C ,  A ,  R
) )  ->  E. v  e.  ( F " C
) w S v ) )
2625impr 602 . 2  |-  ( (
ph  /\  ( w  e.  B  /\  w S ( F `  sup ( C ,  A ,  R ) ) ) )  ->  E. v  e.  ( F " C
) w S v )
275, 12, 23, 26eqsupd 7208 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    Or wor 4313   "cima 4692   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256   supcsup 7193
This theorem is referenced by:  infmsup  9732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-sup 7194
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