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Theorem djuinj 6959
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 6914 . . . . . . 7  |-  (inl  |`  dom  R
) : dom  R -1-1-> ( dom  R A )
2 f1fun 5301 . . . . . . 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 5152 . . . . . 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 5133 . . . . 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 4694 . . . . 5  |-  `' ( R  o.  `' (inl  |`  dom  R ) )  =  ( `' `' (inl  |`  dom  R )  o.  `' R )
109funeqi 5114 . . . 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 6915 . . . . . . 7  |-  (inr  |`  dom  S
) : dom  S -1-1-> ( A dom  S )
13 f1fun 5301 . . . . . . 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 5152 . . . . . 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 5133 . . . . 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 4694 . . . . 5  |-  `' ( S  o.  `' (inr  |`  dom  S ) )  =  ( `' `' (inr  |`  dom  S )  o.  `' S )
2120funeqi 5114 . . . 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 4520 . . . . . . 7  |-  ran  ( R  o.  `' (inl  |` 
dom  R ) )  =  dom  `' ( R  o.  `' (inl  |`  dom  R ) )
24 rncoss 4779 . . . . . . 7  |-  ran  ( R  o.  `' (inl  |` 
dom  R ) ) 
C_  ran  R
2523, 24eqsstrri 3100 . . . . . 6  |-  dom  `' ( R  o.  `' (inl  |`  dom  R ) )  C_  ran  R
26 df-rn 4520 . . . . . . 7  |-  ran  ( S  o.  `' (inr  |` 
dom  S ) )  =  dom  `' ( S  o.  `' (inr  |`  dom  S ) )
27 rncoss 4779 . . . . . . 7  |-  ran  ( S  o.  `' (inr  |` 
dom  S ) ) 
C_  ran  S
2826, 27eqsstrri 3100 . . . . . 6  |-  dom  `' ( S  o.  `' (inr  |`  dom  S ) )  C_  ran  S
29 ss2in 3274 . . . . . 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 3117 . . . 4  |-  ( ph  ->  ( dom  `' ( R  o.  `' (inl  |`  dom  R ) )  i^i  dom  `' ( S  o.  `' (inr  |` 
dom  S ) ) )  C_  (/) )
33 ss0 3373 . . . 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 5137 . . 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 1200 . 2  |-  ( ph  ->  Fun  ( `' ( R  o.  `' (inl  |`  dom  R ) )  u.  `' ( S  o.  `' (inr  |`  dom  S
) ) ) )
37 df-djud 6956 . . . . 5  |-  ( R ⊔d  S )  =  ( ( R  o.  `' (inl  |`  dom  R ) )  u.  ( S  o.  `' (inr  |`  dom  S
) ) )
3837cnveqi 4684 . . . 4  |-  `' ( R ⊔d  S )  =  `' ( ( R  o.  `' (inl  |`  dom  R
) )  u.  ( S  o.  `' (inr  |` 
dom  S ) ) )
39 cnvun 4914 . . . 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 2138 . . 3  |-  `' ( R ⊔d  S )  =  ( `' ( R  o.  `' (inl  |`  dom  R
) )  u.  `' ( S  o.  `' (inr  |`  dom  S ) ) )
4140funeqi 5114 . 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 1316    u. cun 3039    i^i cin 3040    C_ wss 3041   (/)c0 3333   `'ccnv 4508   dom cdm 4509   ran crn 4510    |` cres 4511    o. ccom 4513   Fun wfun 5087   -1-1->wf1 5090   ⊔ cdju 6890  inlcinl 6898  inrcinr 6899   ⊔d cdjud 6955
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901  df-djud 6956
This theorem is referenced by: (None)
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