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Theorem infisoti 7091
Description: Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
Hypotheses
Ref Expression
infisoti.1  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
infisoti.2  |-  ( ph  ->  C  C_  A )
infisoti.3  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  C  z R y ) ) )
infisoti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
Assertion
Ref Expression
infisoti  |-  ( ph  -> inf ( ( F " C ) ,  B ,  S )  =  ( F ` inf ( C ,  A ,  R ) ) )
Distinct variable groups:    u, A, v, x, y, z    u, B, v, x, y, z   
u, C, v, x, y, z    u, F, v, x, y, z   
u, R, v, x, y, z    u, S, v, x, y, z    ph, u, v, x, y, z

Proof of Theorem infisoti
StepHypRef Expression
1 infisoti.1 . . . 4  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
2 isocnv2 5855 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  <-> 
F  Isom  `' R ,  `' S ( A ,  B ) )
31, 2sylib 122 . . 3  |-  ( ph  ->  F  Isom  `' R ,  `' S ( A ,  B ) )
4 infisoti.2 . . 3  |-  ( ph  ->  C  C_  A )
5 infisoti.3 . . . 4  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  C  z R y ) ) )
65cnvinfex 7077 . . 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 ) ) )
7 infisoti.ti . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
87cnvti 7078 . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u `' R v  /\  -.  v `' R u ) ) )
93, 4, 6, 8supisoti 7069 . 2  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  `' S
)  =  ( F `
 sup ( C ,  A ,  `' R ) ) )
10 df-inf 7044 . 2  |- inf ( ( F " C ) ,  B ,  S
)  =  sup (
( F " C
) ,  B ,  `' S )
11 df-inf 7044 . . 3  |- inf ( C ,  A ,  R
)  =  sup ( C ,  A ,  `' R )
1211fveq2i 5557 . 2  |-  ( F `
inf ( C ,  A ,  R )
)  =  ( F `
 sup ( C ,  A ,  `' R ) )
139, 10, 123eqtr4g 2251 1  |-  ( ph  -> inf ( ( F " C ) ,  B ,  S )  =  ( F ` inf ( C ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2164   A.wral 2472   E.wrex 2473    C_ wss 3153   class class class wbr 4029   `'ccnv 4658   "cima 4662   ` cfv 5254    Isom wiso 5255   supcsup 7041  infcinf 7042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-sup 7043  df-inf 7044
This theorem is referenced by: (None)
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