ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caseinj Unicode version

Theorem caseinj 6940
Description: The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
caseinj.r  |-  ( ph  ->  Fun  `' R )
caseinj.s  |-  ( ph  ->  Fun  `' S )
caseinj.disj  |-  ( ph  ->  ( ran  R  i^i  ran 
S )  =  (/) )
Assertion
Ref Expression
caseinj  |-  ( ph  ->  Fun  `'case ( R ,  S
) )

Proof of Theorem caseinj
StepHypRef Expression
1 df-inl 6898 . . . . . . 7  |- inl  =  ( y  e.  _V  |->  <. (/)
,  y >. )
21funmpt2 5130 . . . . . 6  |-  Fun inl
3 funcnvcnv 5150 . . . . . 6  |-  ( Fun inl  ->  Fun  `' `'inl )
42, 3ax-mp 5 . . . . 5  |-  Fun  `' `'inl
5 caseinj.r . . . . 5  |-  ( ph  ->  Fun  `' R )
6 funco 5131 . . . . 5  |-  ( ( Fun  `' `'inl  /\  Fun  `' R )  ->  Fun  ( `' `'inl  o.  `' R ) )
74, 5, 6sylancr 408 . . . 4  |-  ( ph  ->  Fun  ( `' `'inl  o.  `' R ) )
8 cnvco 4692 . . . . 5  |-  `' ( R  o.  `'inl )  =  ( `' `'inl  o.  `' R )
98funeqi 5112 . . . 4  |-  ( Fun  `' ( R  o.  `'inl )  <->  Fun  ( `' `'inl  o.  `' R ) )
107, 9sylibr 133 . . 3  |-  ( ph  ->  Fun  `' ( R  o.  `'inl ) )
11 df-inr 6899 . . . . . . 7  |- inr  =  ( x  e.  _V  |->  <. 1o ,  x >. )
1211funmpt2 5130 . . . . . 6  |-  Fun inr
13 funcnvcnv 5150 . . . . . 6  |-  ( Fun inr  ->  Fun  `' `'inr )
1412, 13ax-mp 5 . . . . 5  |-  Fun  `' `'inr
15 caseinj.s . . . . 5  |-  ( ph  ->  Fun  `' S )
16 funco 5131 . . . . 5  |-  ( ( Fun  `' `'inr  /\  Fun  `' S )  ->  Fun  ( `' `'inr  o.  `' S ) )
1714, 15, 16sylancr 408 . . . 4  |-  ( ph  ->  Fun  ( `' `'inr  o.  `' S ) )
18 cnvco 4692 . . . . 5  |-  `' ( S  o.  `'inr )  =  ( `' `'inr  o.  `' S )
1918funeqi 5112 . . . 4  |-  ( Fun  `' ( S  o.  `'inr )  <->  Fun  ( `' `'inr  o.  `' S ) )
2017, 19sylibr 133 . . 3  |-  ( ph  ->  Fun  `' ( S  o.  `'inr ) )
21 df-rn 4518 . . . . . . 7  |-  ran  ( R  o.  `'inl )  =  dom  `' ( R  o.  `'inl )
22 rncoss 4777 . . . . . . 7  |-  ran  ( R  o.  `'inl )  C_ 
ran  R
2321, 22eqsstrri 3098 . . . . . 6  |-  dom  `' ( R  o.  `'inl )  C_  ran  R
24 df-rn 4518 . . . . . . 7  |-  ran  ( S  o.  `'inr )  =  dom  `' ( S  o.  `'inr )
25 rncoss 4777 . . . . . . 7  |-  ran  ( S  o.  `'inr )  C_ 
ran  S
2624, 25eqsstrri 3098 . . . . . 6  |-  dom  `' ( S  o.  `'inr )  C_  ran  S
27 ss2in 3272 . . . . . 6  |-  ( ( dom  `' ( R  o.  `'inl )  C_  ran  R  /\  dom  `' ( S  o.  `'inr )  C_  ran  S )  ->  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr )
)  C_  ( ran  R  i^i  ran  S )
)
2823, 26, 27mp2an 420 . . . . 5  |-  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr ) )  C_  ( ran  R  i^i  ran  S
)
29 caseinj.disj . . . . 5  |-  ( ph  ->  ( ran  R  i^i  ran 
S )  =  (/) )
3028, 29sseqtrid 3115 . . . 4  |-  ( ph  ->  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr ) ) 
C_  (/) )
31 ss0 3371 . . . 4  |-  ( ( dom  `' ( R  o.  `'inl )  i^i 
dom  `' ( S  o.  `'inr ) )  C_  (/)  ->  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr ) )  =  (/) )
3230, 31syl 14 . . 3  |-  ( ph  ->  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr ) )  =  (/) )
33 funun 5135 . . 3  |-  ( ( ( Fun  `' ( R  o.  `'inl )  /\  Fun  `' ( S  o.  `'inr ) )  /\  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr )
)  =  (/) )  ->  Fun  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) ) )
3410, 20, 32, 33syl21anc 1198 . 2  |-  ( ph  ->  Fun  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) ) )
35 df-case 6935 . . . . 5  |- case ( R ,  S )  =  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr )
)
3635cnveqi 4682 . . . 4  |-  `'case ( R ,  S )  =  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )
37 cnvun 4912 . . . 4  |-  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3836, 37eqtri 2136 . . 3  |-  `'case ( R ,  S )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3938funeqi 5112 . 2  |-  ( Fun  `'case ( R ,  S
)  <->  Fun  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) ) )
4034, 39sylibr 133 1  |-  ( ph  ->  Fun  `'case ( R ,  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314   _Vcvv 2658    u. cun 3037    i^i cin 3038    C_ wss 3039   (/)c0 3331   <.cop 3498   `'ccnv 4506   dom cdm 4507   ran crn 4508    o. ccom 4511   Fun wfun 5085   1oc1o 6272  inlcinl 6896  inrcinr 6897  casecdjucase 6934
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-inl 6898  df-inr 6899  df-case 6935
This theorem is referenced by:  casef1  6941
  Copyright terms: Public domain W3C validator