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Theorem djuinj 6984
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 6939 . . . . . . 7  |-  (inl  |`  dom  R
) : dom  R -1-1-> ( dom  R A )
2 f1fun 5326 . . . . . . 7  |-  ( (inl  |`  dom  R ) : dom  R -1-1-> ( dom 
R A )  ->  Fun  (inl  |`  dom  R ) )
31, 2ax-mp 5 . . . . . 6  |-  Fun  (inl  |` 
dom  R )
4 funcnvcnv 5177 . . . . . 6  |-  ( Fun  (inl  |`  dom  R )  ->  Fun  `' `' (inl  |`  dom  R ) )
53, 4ax-mp 5 . . . . 5  |-  Fun  `' `' (inl  |`  dom  R
)
6 djuinj.r . . . . 5  |-  ( ph  ->  Fun  `' R )
7 funco 5158 . . . . 5  |-  ( ( Fun  `' `' (inl  |`  dom  R )  /\  Fun  `' R )  ->  Fun  ( `' `' (inl  |`  dom  R
)  o.  `' R
) )
85, 6, 7sylancr 410 . . . 4  |-  ( ph  ->  Fun  ( `' `' (inl  |`  dom  R )  o.  `' R ) )
9 cnvco 4719 . . . . 5  |-  `' ( R  o.  `' (inl  |`  dom  R ) )  =  ( `' `' (inl  |`  dom  R )  o.  `' R )
109funeqi 5139 . . . 4  |-  ( Fun  `' ( R  o.  `' (inl  |`  dom  R
) )  <->  Fun  ( `' `' (inl  |`  dom  R
)  o.  `' R
) )
118, 10sylibr 133 . . 3  |-  ( ph  ->  Fun  `' ( R  o.  `' (inl  |`  dom  R
) ) )
12 inrresf1 6940 . . . . . . 7  |-  (inr  |`  dom  S
) : dom  S -1-1-> ( A dom  S )
13 f1fun 5326 . . . . . . 7  |-  ( (inr  |`  dom  S ) : dom  S -1-1-> ( A dom  S )  ->  Fun  (inr  |`  dom  S ) )
1412, 13ax-mp 5 . . . . . 6  |-  Fun  (inr  |` 
dom  S )
15 funcnvcnv 5177 . . . . . 6  |-  ( Fun  (inr  |`  dom  S )  ->  Fun  `' `' (inr  |`  dom  S ) )
1614, 15ax-mp 5 . . . . 5  |-  Fun  `' `' (inr  |`  dom  S
)
17 djuinj.s . . . . 5  |-  ( ph  ->  Fun  `' S )
18 funco 5158 . . . . 5  |-  ( ( Fun  `' `' (inr  |`  dom  S )  /\  Fun  `' S )  ->  Fun  ( `' `' (inr  |`  dom  S
)  o.  `' S
) )
1916, 17, 18sylancr 410 . . . 4  |-  ( ph  ->  Fun  ( `' `' (inr  |`  dom  S )  o.  `' S ) )
20 cnvco 4719 . . . . 5  |-  `' ( S  o.  `' (inr  |`  dom  S ) )  =  ( `' `' (inr  |`  dom  S )  o.  `' S )
2120funeqi 5139 . . . 4  |-  ( Fun  `' ( S  o.  `' (inr  |`  dom  S
) )  <->  Fun  ( `' `' (inr  |`  dom  S
)  o.  `' S
) )
2219, 21sylibr 133 . . 3  |-  ( ph  ->  Fun  `' ( S  o.  `' (inr  |`  dom  S
) ) )
23 df-rn 4545 . . . . . . 7  |-  ran  ( R  o.  `' (inl  |` 
dom  R ) )  =  dom  `' ( R  o.  `' (inl  |`  dom  R ) )
24 rncoss 4804 . . . . . . 7  |-  ran  ( R  o.  `' (inl  |` 
dom  R ) ) 
C_  ran  R
2523, 24eqsstrri 3125 . . . . . 6  |-  dom  `' ( R  o.  `' (inl  |`  dom  R ) )  C_  ran  R
26 df-rn 4545 . . . . . . 7  |-  ran  ( S  o.  `' (inr  |` 
dom  S ) )  =  dom  `' ( S  o.  `' (inr  |`  dom  S ) )
27 rncoss 4804 . . . . . . 7  |-  ran  ( S  o.  `' (inr  |` 
dom  S ) ) 
C_  ran  S
2826, 27eqsstrri 3125 . . . . . 6  |-  dom  `' ( S  o.  `' (inr  |`  dom  S ) )  C_  ran  S
29 ss2in 3299 . . . . . 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 422 . . . . 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, 31sseqtrid 3142 . . . 4  |-  ( ph  ->  ( dom  `' ( R  o.  `' (inl  |`  dom  R ) )  i^i  dom  `' ( S  o.  `' (inr  |` 
dom  S ) ) )  C_  (/) )
33 ss0 3398 . . . 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 5162 . . 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 1215 . 2  |-  ( ph  ->  Fun  ( `' ( R  o.  `' (inl  |`  dom  R ) )  u.  `' ( S  o.  `' (inr  |`  dom  S
) ) ) )
37 df-djud 6981 . . . . 5  |-  ( R ⊔d  S )  =  ( ( R  o.  `' (inl  |`  dom  R ) )  u.  ( S  o.  `' (inr  |`  dom  S
) ) )
3837cnveqi 4709 . . . 4  |-  `' ( R ⊔d  S )  =  `' ( ( R  o.  `' (inl  |`  dom  R
) )  u.  ( S  o.  `' (inr  |` 
dom  S ) ) )
39 cnvun 4939 . . . 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 2158 . . 3  |-  `' ( R ⊔d  S )  =  ( `' ( R  o.  `' (inl  |`  dom  R
) )  u.  `' ( S  o.  `' (inr  |`  dom  S ) ) )
4140funeqi 5139 . 2  |-  ( Fun  `' ( R ⊔d  S )  <->  Fun  ( `' ( R  o.  `' (inl  |`  dom  R
) )  u.  `' ( S  o.  `' (inr  |`  dom  S ) ) ) )
4236, 41sylibr 133 1  |-  ( ph  ->  Fun  `' ( R ⊔d  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    u. cun 3064    i^i cin 3065    C_ wss 3066   (/)c0 3358   `'ccnv 4533   dom cdm 4534   ran crn 4535    |` cres 4536    o. ccom 4538   Fun wfun 5112   -1-1->wf1 5115   ⊔ cdju 6915  inlcinl 6923  inrcinr 6924   ⊔d cdjud 6980
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-dju 6916  df-inl 6925  df-inr 6926  df-djud 6981
This theorem is referenced by: (None)
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