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Theorem caseinj 6778
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 6737 . . . . . . 7  |- inl  =  ( y  e.  _V  |->  <. (/)
,  y >. )
21funmpt2 5053 . . . . . 6  |-  Fun inl
3 funcnvcnv 5073 . . . . . 6  |-  ( Fun inl  ->  Fun  `' `'inl )
42, 3ax-mp 7 . . . . 5  |-  Fun  `' `'inl
5 caseinj.r . . . . 5  |-  ( ph  ->  Fun  `' R )
6 funco 5054 . . . . 5  |-  ( ( Fun  `' `'inl  /\  Fun  `' R )  ->  Fun  ( `' `'inl  o.  `' R ) )
74, 5, 6sylancr 405 . . . 4  |-  ( ph  ->  Fun  ( `' `'inl  o.  `' R ) )
8 cnvco 4621 . . . . 5  |-  `' ( R  o.  `'inl )  =  ( `' `'inl  o.  `' R )
98funeqi 5036 . . . 4  |-  ( Fun  `' ( R  o.  `'inl )  <->  Fun  ( `' `'inl  o.  `' R ) )
107, 9sylibr 132 . . 3  |-  ( ph  ->  Fun  `' ( R  o.  `'inl ) )
11 df-inr 6738 . . . . . . 7  |- inr  =  ( x  e.  _V  |->  <. 1o ,  x >. )
1211funmpt2 5053 . . . . . 6  |-  Fun inr
13 funcnvcnv 5073 . . . . . 6  |-  ( Fun inr  ->  Fun  `' `'inr )
1412, 13ax-mp 7 . . . . 5  |-  Fun  `' `'inr
15 caseinj.s . . . . 5  |-  ( ph  ->  Fun  `' S )
16 funco 5054 . . . . 5  |-  ( ( Fun  `' `'inr  /\  Fun  `' S )  ->  Fun  ( `' `'inr  o.  `' S ) )
1714, 15, 16sylancr 405 . . . 4  |-  ( ph  ->  Fun  ( `' `'inr  o.  `' S ) )
18 cnvco 4621 . . . . 5  |-  `' ( S  o.  `'inr )  =  ( `' `'inr  o.  `' S )
1918funeqi 5036 . . . 4  |-  ( Fun  `' ( S  o.  `'inr )  <->  Fun  ( `' `'inr  o.  `' S ) )
2017, 19sylibr 132 . . 3  |-  ( ph  ->  Fun  `' ( S  o.  `'inr ) )
21 df-rn 4449 . . . . . . 7  |-  ran  ( R  o.  `'inl )  =  dom  `' ( R  o.  `'inl )
22 rncoss 4703 . . . . . . 7  |-  ran  ( R  o.  `'inl )  C_ 
ran  R
2321, 22eqsstr3i 3057 . . . . . 6  |-  dom  `' ( R  o.  `'inl )  C_  ran  R
24 df-rn 4449 . . . . . . 7  |-  ran  ( S  o.  `'inr )  =  dom  `' ( S  o.  `'inr )
25 rncoss 4703 . . . . . . 7  |-  ran  ( S  o.  `'inr )  C_ 
ran  S
2624, 25eqsstr3i 3057 . . . . . 6  |-  dom  `' ( S  o.  `'inr )  C_  ran  S
27 ss2in 3227 . . . . . 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 417 . . . . 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, 29syl5sseq 3074 . . . 4  |-  ( ph  ->  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr ) ) 
C_  (/) )
31 ss0 3323 . . . 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 5058 . . 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 1173 . 2  |-  ( ph  ->  Fun  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) ) )
35 df-case 6773 . . . . 5  |- case ( R ,  S )  =  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr )
)
3635cnveqi 4611 . . . 4  |-  `'case ( R ,  S )  =  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )
37 cnvun 4837 . . . 4  |-  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3836, 37eqtri 2108 . . 3  |-  `'case ( R ,  S )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3938funeqi 5036 . 2  |-  ( Fun  `'case ( R ,  S
)  <->  Fun  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) ) )
4034, 39sylibr 132 1  |-  ( ph  ->  Fun  `'case ( R ,  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   _Vcvv 2619    u. cun 2997    i^i cin 2998    C_ wss 2999   (/)c0 3286   <.cop 3449   `'ccnv 4437   dom cdm 4438   ran crn 4439    o. ccom 4442   Fun wfun 5009   1oc1o 6174  inlcinl 6735  inrcinr 6736  casecdjucase 6772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-fun 5017  df-inl 6737  df-inr 6738  df-case 6773
This theorem is referenced by:  casef1  6779
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