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Theorem caserel 6940
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 6937 . 2  |- case ( R ,  S )  =  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr )
)
2 cocnvss 5034 . . . 4  |-  ( R  o.  `'inl )  C_  ( ran  (inl  |`  dom  R
)  X.  ran  ( R  |`  dom inl ) )
3 inlresf1 6914 . . . . . 6  |-  (inl  |`  dom  R
) : dom  R -1-1-> ( dom  R dom  S )
4 f1rn 5299 . . . . . 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 4813 . . . . . . 7  |-  ( R  |`  dom inl )  C_  R
7 rnss 4739 . . . . . . 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 3209 . . . . . 6  |-  ran  R  C_  ( ran  R  u.  ran  S )
108, 9sstri 3076 . . . . 5  |-  ran  ( R  |`  dom inl )  C_  ( ran  R  u.  ran  S
)
11 xpss12 4616 . . . . 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 422 . . . 4  |-  ( ran  (inl  |`  dom  R )  X.  ran  ( R  |`  dom inl ) )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )
132, 12sstri 3076 . . 3  |-  ( R  o.  `'inl )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )
14 cocnvss 5034 . . . 4  |-  ( S  o.  `'inr )  C_  ( ran  (inr  |`  dom  S
)  X.  ran  ( S  |`  dom inr ) )
15 inrresf1 6915 . . . . . 6  |-  (inr  |`  dom  S
) : dom  S -1-1-> ( dom  R dom  S )
16 f1rn 5299 . . . . . 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 4813 . . . . . . 7  |-  ( S  |`  dom inr )  C_  S
19 rnss 4739 . . . . . . 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 3210 . . . . . 6  |-  ran  S  C_  ( ran  R  u.  ran  S )
2220, 21sstri 3076 . . . . 5  |-  ran  ( S  |`  dom inr )  C_  ( ran  R  u.  ran  S
)
23 xpss12 4616 . . . . 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 422 . . . 4  |-  ( ran  (inr  |`  dom  S )  X.  ran  ( S  |`  dom inr ) )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )
2514, 24sstri 3076 . . 3  |-  ( S  o.  `'inr )  C_  ( ( dom  R dom 
S )  X.  ( ran  R  u.  ran  S
) )
2613, 25unssi 3221 . 2  |-  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )  C_  (
( dom  R dom  S
)  X.  ( ran 
R  u.  ran  S
) )
271, 26eqsstri 3099 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 3039    C_ wss 3041    X. cxp 4507   `'ccnv 4508   dom cdm 4509   ran crn 4510    |` cres 4511    o. ccom 4513   -1-1->wf1 5090   ⊔ cdju 6890  inlcinl 6898  inrcinr 6899  casecdjucase 6936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901  df-case 6937
This theorem is referenced by:  casef  6941
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