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Theorem caserel 7117
Description: The "case" construction of two relations is a relation, with bounds on its domain and codomain. Typically, the "case" construction is used when both relations have a common codomain. (Contributed by BJ, 10-Jul-2022.)
Assertion
Ref Expression
caserel  |- case ( R ,  S )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )

Proof of Theorem caserel
StepHypRef Expression
1 df-case 7114 . 2  |- case ( R ,  S )  =  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr )
)
2 cocnvss 5172 . . . 4  |-  ( R  o.  `'inl )  C_  ( ran  (inl  |`  dom  R
)  X.  ran  ( R  |`  dom inl ) )
3 inlresf1 7091 . . . . . 6  |-  (inl  |`  dom  R
) : dom  R -1-1-> ( dom  R dom  S )
4 f1rn 5441 . . . . . 6  |-  ( (inl  |`  dom  R ) : dom  R -1-1-> ( dom 
R dom  S )  ->  ran  (inl  |`  dom  R ) 
C_  ( dom  R dom 
S ) )
53, 4ax-mp 5 . . . . 5  |-  ran  (inl  |` 
dom  R )  C_  ( dom  R dom  S )
6 resss 4949 . . . . . . 7  |-  ( R  |`  dom inl )  C_  R
7 rnss 4875 . . . . . . 7  |-  ( ( R  |`  dom inl )  C_  R  ->  ran  ( R  |` 
dom inl )  C_  ran  R
)
86, 7ax-mp 5 . . . . . 6  |-  ran  ( R  |`  dom inl )  C_  ran  R
9 ssun1 3313 . . . . . 6  |-  ran  R  C_  ( ran  R  u.  ran  S )
108, 9sstri 3179 . . . . 5  |-  ran  ( R  |`  dom inl )  C_  ( ran  R  u.  ran  S
)
11 xpss12 4751 . . . . 5  |-  ( ( ran  (inl  |`  dom  R
)  C_  ( dom  R dom  S )  /\  ran  ( R  |`  dom inl )  C_  ( ran  R  u.  ran  S ) )  -> 
( ran  (inl  |`  dom  R
)  X.  ran  ( R  |`  dom inl ) )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) ) )
125, 10, 11mp2an 426 . . . 4  |-  ( ran  (inl  |`  dom  R )  X.  ran  ( R  |`  dom inl ) )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )
132, 12sstri 3179 . . 3  |-  ( R  o.  `'inl )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )
14 cocnvss 5172 . . . 4  |-  ( S  o.  `'inr )  C_  ( ran  (inr  |`  dom  S
)  X.  ran  ( S  |`  dom inr ) )
15 inrresf1 7092 . . . . . 6  |-  (inr  |`  dom  S
) : dom  S -1-1-> ( dom  R dom  S )
16 f1rn 5441 . . . . . 6  |-  ( (inr  |`  dom  S ) : dom  S -1-1-> ( dom 
R dom  S )  ->  ran  (inr  |`  dom  S ) 
C_  ( dom  R dom 
S ) )
1715, 16ax-mp 5 . . . . 5  |-  ran  (inr  |` 
dom  S )  C_  ( dom  R dom  S )
18 resss 4949 . . . . . . 7  |-  ( S  |`  dom inr )  C_  S
19 rnss 4875 . . . . . . 7  |-  ( ( S  |`  dom inr )  C_  S  ->  ran  ( S  |` 
dom inr )  C_  ran  S
)
2018, 19ax-mp 5 . . . . . 6  |-  ran  ( S  |`  dom inr )  C_  ran  S
21 ssun2 3314 . . . . . 6  |-  ran  S  C_  ( ran  R  u.  ran  S )
2220, 21sstri 3179 . . . . 5  |-  ran  ( S  |`  dom inr )  C_  ( ran  R  u.  ran  S
)
23 xpss12 4751 . . . . 5  |-  ( ( ran  (inr  |`  dom  S
)  C_  ( dom  R dom  S )  /\  ran  ( S  |`  dom inr )  C_  ( ran  R  u.  ran  S ) )  -> 
( ran  (inr  |`  dom  S
)  X.  ran  ( S  |`  dom inr ) )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) ) )
2417, 22, 23mp2an 426 . . . 4  |-  ( ran  (inr  |`  dom  S )  X.  ran  ( S  |`  dom inr ) )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )
2514, 24sstri 3179 . . 3  |-  ( S  o.  `'inr )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )
2613, 25unssi 3325 . 2  |-  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )  C_  (
( dom  R dom  S
)  X.  ( ran 
R  u.  ran  S
) )
271, 26eqsstri 3202 1  |- case ( R ,  S )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )
Colors of variables: wff set class
Syntax hints:    u. cun 3142    C_ wss 3144    X. cxp 4642   `'ccnv 4643   dom cdm 4644   ran crn 4645    |` cres 4646    o. ccom 4648   -1-1->wf1 5232   ⊔ cdju 7067  inlcinl 7075  inrcinr 7076  casecdjucase 7113
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-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6166  df-2nd 6167  df-1o 6442  df-dju 7068  df-inl 7077  df-inr 7078  df-case 7114
This theorem is referenced by:  casef  7118
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