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Theorem caseinj 7087
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 7045 . . . . . . 7  |- inl  =  ( y  e.  _V  |->  <. (/)
,  y >. )
21funmpt2 5255 . . . . . 6  |-  Fun inl
3 funcnvcnv 5275 . . . . . 6  |-  ( Fun inl  ->  Fun  `' `'inl )
42, 3ax-mp 5 . . . . 5  |-  Fun  `' `'inl
5 caseinj.r . . . . 5  |-  ( ph  ->  Fun  `' R )
6 funco 5256 . . . . 5  |-  ( ( Fun  `' `'inl  /\  Fun  `' R )  ->  Fun  ( `' `'inl  o.  `' R ) )
74, 5, 6sylancr 414 . . . 4  |-  ( ph  ->  Fun  ( `' `'inl  o.  `' R ) )
8 cnvco 4812 . . . . 5  |-  `' ( R  o.  `'inl )  =  ( `' `'inl  o.  `' R )
98funeqi 5237 . . . 4  |-  ( Fun  `' ( R  o.  `'inl )  <->  Fun  ( `' `'inl  o.  `' R ) )
107, 9sylibr 134 . . 3  |-  ( ph  ->  Fun  `' ( R  o.  `'inl ) )
11 df-inr 7046 . . . . . . 7  |- inr  =  ( x  e.  _V  |->  <. 1o ,  x >. )
1211funmpt2 5255 . . . . . 6  |-  Fun inr
13 funcnvcnv 5275 . . . . . 6  |-  ( Fun inr  ->  Fun  `' `'inr )
1412, 13ax-mp 5 . . . . 5  |-  Fun  `' `'inr
15 caseinj.s . . . . 5  |-  ( ph  ->  Fun  `' S )
16 funco 5256 . . . . 5  |-  ( ( Fun  `' `'inr  /\  Fun  `' S )  ->  Fun  ( `' `'inr  o.  `' S ) )
1714, 15, 16sylancr 414 . . . 4  |-  ( ph  ->  Fun  ( `' `'inr  o.  `' S ) )
18 cnvco 4812 . . . . 5  |-  `' ( S  o.  `'inr )  =  ( `' `'inr  o.  `' S )
1918funeqi 5237 . . . 4  |-  ( Fun  `' ( S  o.  `'inr )  <->  Fun  ( `' `'inr  o.  `' S ) )
2017, 19sylibr 134 . . 3  |-  ( ph  ->  Fun  `' ( S  o.  `'inr ) )
21 df-rn 4637 . . . . . . 7  |-  ran  ( R  o.  `'inl )  =  dom  `' ( R  o.  `'inl )
22 rncoss 4897 . . . . . . 7  |-  ran  ( R  o.  `'inl )  C_ 
ran  R
2321, 22eqsstrri 3188 . . . . . 6  |-  dom  `' ( R  o.  `'inl )  C_  ran  R
24 df-rn 4637 . . . . . . 7  |-  ran  ( S  o.  `'inr )  =  dom  `' ( S  o.  `'inr )
25 rncoss 4897 . . . . . . 7  |-  ran  ( S  o.  `'inr )  C_ 
ran  S
2624, 25eqsstrri 3188 . . . . . 6  |-  dom  `' ( S  o.  `'inr )  C_  ran  S
27 ss2in 3363 . . . . . 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 426 . . . . 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 3205 . . . 4  |-  ( ph  ->  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr ) ) 
C_  (/) )
31 ss0 3463 . . . 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 5260 . . 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 1237 . 2  |-  ( ph  ->  Fun  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) ) )
35 df-case 7082 . . . . 5  |- case ( R ,  S )  =  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr )
)
3635cnveqi 4802 . . . 4  |-  `'case ( R ,  S )  =  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )
37 cnvun 5034 . . . 4  |-  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3836, 37eqtri 2198 . . 3  |-  `'case ( R ,  S )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3938funeqi 5237 . 2  |-  ( Fun  `'case ( R ,  S
)  <->  Fun  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) ) )
4034, 39sylibr 134 1  |-  ( ph  ->  Fun  `'case ( R ,  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   _Vcvv 2737    u. cun 3127    i^i cin 3128    C_ wss 3129   (/)c0 3422   <.cop 3595   `'ccnv 4625   dom cdm 4626   ran crn 4627    o. ccom 4630   Fun wfun 5210   1oc1o 6409  inlcinl 7043  inrcinr 7044  casecdjucase 7081
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-inl 7045  df-inr 7046  df-case 7082
This theorem is referenced by:  casef1  7088
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