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

Theorem djuinj 6784
Description: The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
djuinj.r  |-  ( ph  ->  Fun  `' R )
djuinj.s  |-  ( ph  ->  Fun  `' S )
djuinj.disj  |-  ( ph  ->  ( ran  R  i^i  ran 
S )  =  (/) )
Assertion
Ref Expression
djuinj  |-  ( ph  ->  Fun  `' ( R ⊔d  S ) )

Proof of Theorem djuinj
StepHypRef Expression
1 inlresf1 6751 . . . . . . 7  |-  (inl  |`  dom  R
) : dom  R -1-1-> ( dom  R A )
2 f1fun 5219 . . . . . . 7  |-  ( (inl  |`  dom  R ) : dom  R -1-1-> ( dom 
R A )  ->  Fun  (inl  |`  dom  R ) )
31, 2ax-mp 7 . . . . . 6  |-  Fun  (inl  |` 
dom  R )
4 funcnvcnv 5073 . . . . . 6  |-  ( Fun  (inl  |`  dom  R )  ->  Fun  `' `' (inl  |`  dom  R ) )
53, 4ax-mp 7 . . . . 5  |-  Fun  `' `' (inl  |`  dom  R
)
6 djuinj.r . . . . 5  |-  ( ph  ->  Fun  `' R )
7 funco 5054 . . . . 5  |-  ( ( Fun  `' `' (inl  |`  dom  R )  /\  Fun  `' R )  ->  Fun  ( `' `' (inl  |`  dom  R
)  o.  `' R
) )
85, 6, 7sylancr 405 . . . 4  |-  ( ph  ->  Fun  ( `' `' (inl  |`  dom  R )  o.  `' R ) )
9 cnvco 4621 . . . . 5  |-  `' ( R  o.  `' (inl  |`  dom  R ) )  =  ( `' `' (inl  |`  dom  R )  o.  `' R )
109funeqi 5036 . . . 4  |-  ( Fun  `' ( R  o.  `' (inl  |`  dom  R
) )  <->  Fun  ( `' `' (inl  |`  dom  R
)  o.  `' R
) )
118, 10sylibr 132 . . 3  |-  ( ph  ->  Fun  `' ( R  o.  `' (inl  |`  dom  R
) ) )
12 inrresf1 6752 . . . . . . 7  |-  (inr  |`  dom  S
) : dom  S -1-1-> ( A dom  S )
13 f1fun 5219 . . . . . . 7  |-  ( (inr  |`  dom  S ) : dom  S -1-1-> ( A dom  S )  ->  Fun  (inr  |`  dom  S ) )
1412, 13ax-mp 7 . . . . . 6  |-  Fun  (inr  |` 
dom  S )
15 funcnvcnv 5073 . . . . . 6  |-  ( Fun  (inr  |`  dom  S )  ->  Fun  `' `' (inr  |`  dom  S ) )
1614, 15ax-mp 7 . . . . 5  |-  Fun  `' `' (inr  |`  dom  S
)
17 djuinj.s . . . . 5  |-  ( ph  ->  Fun  `' S )
18 funco 5054 . . . . 5  |-  ( ( Fun  `' `' (inr  |`  dom  S )  /\  Fun  `' S )  ->  Fun  ( `' `' (inr  |`  dom  S
)  o.  `' S
) )
1916, 17, 18sylancr 405 . . . 4  |-  ( ph  ->  Fun  ( `' `' (inr  |`  dom  S )  o.  `' S ) )
20 cnvco 4621 . . . . 5  |-  `' ( S  o.  `' (inr  |`  dom  S ) )  =  ( `' `' (inr  |`  dom  S )  o.  `' S )
2120funeqi 5036 . . . 4  |-  ( Fun  `' ( S  o.  `' (inr  |`  dom  S
) )  <->  Fun  ( `' `' (inr  |`  dom  S
)  o.  `' S
) )
2219, 21sylibr 132 . . 3  |-  ( ph  ->  Fun  `' ( S  o.  `' (inr  |`  dom  S
) ) )
23 df-rn 4449 . . . . . . 7  |-  ran  ( R  o.  `' (inl  |` 
dom  R ) )  =  dom  `' ( R  o.  `' (inl  |`  dom  R ) )
24 rncoss 4703 . . . . . . 7  |-  ran  ( R  o.  `' (inl  |` 
dom  R ) ) 
C_  ran  R
2523, 24eqsstr3i 3057 . . . . . 6  |-  dom  `' ( R  o.  `' (inl  |`  dom  R ) )  C_  ran  R
26 df-rn 4449 . . . . . . 7  |-  ran  ( S  o.  `' (inr  |` 
dom  S ) )  =  dom  `' ( S  o.  `' (inr  |`  dom  S ) )
27 rncoss 4703 . . . . . . 7  |-  ran  ( S  o.  `' (inr  |` 
dom  S ) ) 
C_  ran  S
2826, 27eqsstr3i 3057 . . . . . 6  |-  dom  `' ( S  o.  `' (inr  |`  dom  S ) )  C_  ran  S
29 ss2in 3227 . . . . . 6  |-  ( ( dom  `' ( R  o.  `' (inl  |`  dom  R
) )  C_  ran  R  /\  dom  `' ( S  o.  `' (inr  |`  dom  S ) ) 
C_  ran  S )  ->  ( dom  `' ( R  o.  `' (inl  |`  dom  R ) )  i^i  dom  `' ( S  o.  `' (inr  |` 
dom  S ) ) )  C_  ( ran  R  i^i  ran  S )
)
3025, 28, 29mp2an 417 . . . . 5  |-  ( dom  `' ( R  o.  `' (inl  |`  dom  R
) )  i^i  dom  `' ( S  o.  `' (inr  |`  dom  S ) ) )  C_  ( ran  R  i^i  ran  S
)
31 djuinj.disj . . . . 5  |-  ( ph  ->  ( ran  R  i^i  ran 
S )  =  (/) )
3230, 31syl5sseq 3074 . . . 4  |-  ( ph  ->  ( dom  `' ( R  o.  `' (inl  |`  dom  R ) )  i^i  dom  `' ( S  o.  `' (inr  |` 
dom  S ) ) )  C_  (/) )
33 ss0 3323 . . . 4  |-  ( ( dom  `' ( R  o.  `' (inl  |`  dom  R
) )  i^i  dom  `' ( S  o.  `' (inr  |`  dom  S ) ) )  C_  (/)  ->  ( dom  `' ( R  o.  `' (inl  |`  dom  R
) )  i^i  dom  `' ( S  o.  `' (inr  |`  dom  S ) ) )  =  (/) )
3432, 33syl 14 . . 3  |-  ( ph  ->  ( dom  `' ( R  o.  `' (inl  |`  dom  R ) )  i^i  dom  `' ( S  o.  `' (inr  |` 
dom  S ) ) )  =  (/) )
35 funun 5058 . . 3  |-  ( ( ( Fun  `' ( R  o.  `' (inl  |`  dom  R ) )  /\  Fun  `' ( S  o.  `' (inr  |`  dom  S ) ) )  /\  ( dom  `' ( R  o.  `' (inl  |`  dom  R
) )  i^i  dom  `' ( S  o.  `' (inr  |`  dom  S ) ) )  =  (/) )  ->  Fun  ( `' ( R  o.  `' (inl  |`  dom  R ) )  u.  `' ( S  o.  `' (inr  |`  dom  S ) ) ) )
3611, 22, 34, 35syl21anc 1173 . 2  |-  ( ph  ->  Fun  ( `' ( R  o.  `' (inl  |`  dom  R ) )  u.  `' ( S  o.  `' (inr  |`  dom  S
) ) ) )
37 df-djud 6781 . . . . 5  |-  ( R ⊔d  S )  =  ( ( R  o.  `' (inl  |`  dom  R ) )  u.  ( S  o.  `' (inr  |`  dom  S
) ) )
3837cnveqi 4611 . . . 4  |-  `' ( R ⊔d  S )  =  `' ( ( R  o.  `' (inl  |`  dom  R
) )  u.  ( S  o.  `' (inr  |` 
dom  S ) ) )
39 cnvun 4837 . . . 4  |-  `' ( ( R  o.  `' (inl  |`  dom  R ) )  u.  ( S  o.  `' (inr  |`  dom  S
) ) )  =  ( `' ( R  o.  `' (inl  |`  dom  R
) )  u.  `' ( S  o.  `' (inr  |`  dom  S ) ) )
4038, 39eqtri 2108 . . 3  |-  `' ( R ⊔d  S )  =  ( `' ( R  o.  `' (inl  |`  dom  R
) )  u.  `' ( S  o.  `' (inr  |`  dom  S ) ) )
4140funeqi 5036 . 2  |-  ( Fun  `' ( R ⊔d  S )  <->  Fun  ( `' ( R  o.  `' (inl  |`  dom  R
) )  u.  `' ( S  o.  `' (inr  |`  dom  S ) ) ) )
4236, 41sylibr 132 1  |-  ( ph  ->  Fun  `' ( R ⊔d  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    u. cun 2997    i^i cin 2998    C_ wss 2999   (/)c0 3286   `'ccnv 4437   dom cdm 4438   ran crn 4439    |` cres 4440    o. ccom 4442   Fun wfun 5009   -1-1->wf1 5012   ⊔ cdju 6728  inlcinl 6735  inrcinr 6736   ⊔d cdjud 6780
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
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-sbc 2841  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-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-dju 6729  df-inl 6737  df-inr 6738  df-djud 6781
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator