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Theorem caseinj 7066
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 7024 . . . . . . 7  |- inl  =  ( y  e.  _V  |->  <. (/)
,  y >. )
21funmpt2 5237 . . . . . 6  |-  Fun inl
3 funcnvcnv 5257 . . . . . 6  |-  ( Fun inl  ->  Fun  `' `'inl )
42, 3ax-mp 5 . . . . 5  |-  Fun  `' `'inl
5 caseinj.r . . . . 5  |-  ( ph  ->  Fun  `' R )
6 funco 5238 . . . . 5  |-  ( ( Fun  `' `'inl  /\  Fun  `' R )  ->  Fun  ( `' `'inl  o.  `' R ) )
74, 5, 6sylancr 412 . . . 4  |-  ( ph  ->  Fun  ( `' `'inl  o.  `' R ) )
8 cnvco 4796 . . . . 5  |-  `' ( R  o.  `'inl )  =  ( `' `'inl  o.  `' R )
98funeqi 5219 . . . 4  |-  ( Fun  `' ( R  o.  `'inl )  <->  Fun  ( `' `'inl  o.  `' R ) )
107, 9sylibr 133 . . 3  |-  ( ph  ->  Fun  `' ( R  o.  `'inl ) )
11 df-inr 7025 . . . . . . 7  |- inr  =  ( x  e.  _V  |->  <. 1o ,  x >. )
1211funmpt2 5237 . . . . . 6  |-  Fun inr
13 funcnvcnv 5257 . . . . . 6  |-  ( Fun inr  ->  Fun  `' `'inr )
1412, 13ax-mp 5 . . . . 5  |-  Fun  `' `'inr
15 caseinj.s . . . . 5  |-  ( ph  ->  Fun  `' S )
16 funco 5238 . . . . 5  |-  ( ( Fun  `' `'inr  /\  Fun  `' S )  ->  Fun  ( `' `'inr  o.  `' S ) )
1714, 15, 16sylancr 412 . . . 4  |-  ( ph  ->  Fun  ( `' `'inr  o.  `' S ) )
18 cnvco 4796 . . . . 5  |-  `' ( S  o.  `'inr )  =  ( `' `'inr  o.  `' S )
1918funeqi 5219 . . . 4  |-  ( Fun  `' ( S  o.  `'inr )  <->  Fun  ( `' `'inr  o.  `' S ) )
2017, 19sylibr 133 . . 3  |-  ( ph  ->  Fun  `' ( S  o.  `'inr ) )
21 df-rn 4622 . . . . . . 7  |-  ran  ( R  o.  `'inl )  =  dom  `' ( R  o.  `'inl )
22 rncoss 4881 . . . . . . 7  |-  ran  ( R  o.  `'inl )  C_ 
ran  R
2321, 22eqsstrri 3180 . . . . . 6  |-  dom  `' ( R  o.  `'inl )  C_  ran  R
24 df-rn 4622 . . . . . . 7  |-  ran  ( S  o.  `'inr )  =  dom  `' ( S  o.  `'inr )
25 rncoss 4881 . . . . . . 7  |-  ran  ( S  o.  `'inr )  C_ 
ran  S
2624, 25eqsstrri 3180 . . . . . 6  |-  dom  `' ( S  o.  `'inr )  C_  ran  S
27 ss2in 3355 . . . . . 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 424 . . . . 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 3197 . . . 4  |-  ( ph  ->  ( dom  `' ( R  o.  `'inl )  i^i  dom  `' ( S  o.  `'inr ) ) 
C_  (/) )
31 ss0 3455 . . . 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 5242 . . 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 1232 . 2  |-  ( ph  ->  Fun  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) ) )
35 df-case 7061 . . . . 5  |- case ( R ,  S )  =  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr )
)
3635cnveqi 4786 . . . 4  |-  `'case ( R ,  S )  =  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )
37 cnvun 5016 . . . 4  |-  `' ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3836, 37eqtri 2191 . . 3  |-  `'case ( R ,  S )  =  ( `' ( R  o.  `'inl )  u.  `' ( S  o.  `'inr ) )
3938funeqi 5219 . 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 1348   _Vcvv 2730    u. cun 3119    i^i cin 3120    C_ wss 3121   (/)c0 3414   <.cop 3586   `'ccnv 4610   dom cdm 4611   ran crn 4612    o. ccom 4615   Fun wfun 5192   1oc1o 6388  inlcinl 7022  inrcinr 7023  casecdjucase 7060
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-inl 7024  df-inr 7025  df-case 7061
This theorem is referenced by:  casef1  7067
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