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Theorem caseinj 7036
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 6994 . . . . . . 7  |- inl  =  ( y  e.  _V  |->  <. (/)
,  y >. )
21funmpt2 5212 . . . . . 6  |-  Fun inl
3 funcnvcnv 5232 . . . . . 6  |-  ( Fun inl  ->  Fun  `' `'inl )
42, 3ax-mp 5 . . . . 5  |-  Fun  `' `'inl
5 caseinj.r . . . . 5  |-  ( ph  ->  Fun  `' R )
6 funco 5213 . . . . 5  |-  ( ( Fun  `' `'inl  /\  Fun  `' R )  ->  Fun  ( `' `'inl  o.  `' R ) )
74, 5, 6sylancr 411 . . . 4  |-  ( ph  ->  Fun  ( `' `'inl  o.  `' R ) )
8 cnvco 4774 . . . . 5  |-  `' ( R  o.  `'inl )  =  ( `' `'inl  o.  `' R )
98funeqi 5194 . . . 4  |-  ( Fun  `' ( R  o.  `'inl )  <->  Fun  ( `' `'inl  o.  `' R ) )
107, 9sylibr 133 . . 3  |-  ( ph  ->  Fun  `' ( R  o.  `'inl ) )
11 df-inr 6995 . . . . . . 7  |- inr  =  ( x  e.  _V  |->  <. 1o ,  x >. )
1211funmpt2 5212 . . . . . 6  |-  Fun inr
13 funcnvcnv 5232 . . . . . 6  |-  ( Fun inr  ->  Fun  `' `'inr )
1412, 13ax-mp 5 . . . . 5  |-  Fun  `' `'inr
15 caseinj.s . . . . 5  |-  ( ph  ->  Fun  `' S )
16 funco 5213 . . . . 5  |-  ( ( Fun  `' `'inr  /\  Fun  `' S )  ->  Fun  ( `' `'inr  o.  `' S ) )
1714, 15, 16sylancr 411 . . . 4  |-  ( ph  ->  Fun  ( `' `'inr  o.  `' S ) )
18 cnvco 4774 . . . . 5  |-  `' ( S  o.  `'inr )  =  ( `' `'inr  o.  `' S )
1918funeqi 5194 . . . 4  |-  ( Fun  `' ( S  o.  `'inr )  <->  Fun  ( `' `'inr  o.  `' S ) )
2017, 19sylibr 133 . . 3  |-  ( ph  ->  Fun  `' ( S  o.  `'inr ) )
21 df-rn 4600 . . . . . . 7  |-  ran  ( R  o.  `'inl )  =  dom  `' ( R  o.  `'inl )
22 rncoss 4859 . . . . . . 7  |-  ran  ( R  o.  `'inl )  C_ 
ran  R
2321, 22eqsstrri 3161 . . . . . 6  |-  dom  `' ( R  o.  `'inl )  C_  ran  R
24 df-rn 4600 . . . . . . 7  |-  ran  ( S  o.  `'inr )  =  dom  `' ( S  o.  `'inr )
25 rncoss 4859 . . . . . . 7  |-  ran  ( S  o.  `'inr )  C_ 
ran  S
2624, 25eqsstrri 3161 . . . . . 6  |-  dom  `' ( S  o.  `'inr )  C_  ran  S
27 ss2in 3336 . . . . . 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 423 . . . . 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 3178 . . . 4  |-  ( ph  ->  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr ) ) 
C_  (/) )
31 ss0 3435 . . . 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 5217 . . 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 1219 . 2  |-  ( ph  ->  Fun  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) ) )
35 df-case 7031 . . . . 5  |- case ( R ,  S )  =  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr )
)
3635cnveqi 4764 . . . 4  |-  `'case ( R ,  S )  =  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )
37 cnvun 4994 . . . 4  |-  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3836, 37eqtri 2178 . . 3  |-  `'case ( R ,  S )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3938funeqi 5194 . 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 1335   _Vcvv 2712    u. cun 3100    i^i cin 3101    C_ wss 3102   (/)c0 3395   <.cop 3564   `'ccnv 4588   dom cdm 4589   ran crn 4590    o. ccom 4593   Fun wfun 5167   1oc1o 6359  inlcinl 6992  inrcinr 6993  casecdjucase 7030
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4029  df-mpt 4030  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-fun 5175  df-inl 6994  df-inr 6995  df-case 7031
This theorem is referenced by:  casef1  7037
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