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Theorem infisoti 7025
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 5807 . . . 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 7011 . . 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 7012 . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u `' R v  /\  -.  v `' R u ) ) )
93, 4, 6, 8supisoti 7003 . 2  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  `' S
)  =  ( F `
 sup ( C ,  A ,  `' R ) ) )
10 df-inf 6978 . 2  |- inf ( ( F " C ) ,  B ,  S
)  =  sup (
( F " C
) ,  B ,  `' S )
11 df-inf 6978 . . 3  |- inf ( C ,  A ,  R
)  =  sup ( C ,  A ,  `' R )
1211fveq2i 5514 . 2  |-  ( F `
inf ( C ,  A ,  R )
)  =  ( F `
 sup ( C ,  A ,  `' R ) )
139, 10, 123eqtr4g 2235 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 1353    e. wcel 2148   A.wral 2455   E.wrex 2456    C_ wss 3129   class class class wbr 4000   `'ccnv 4622   "cima 4626   ` cfv 5212    Isom wiso 5213   supcsup 6975  infcinf 6976
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-sup 6977  df-inf 6978
This theorem is referenced by: (None)
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