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Theorem djufun 6982
Description: The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
djufun.f  |-  ( ph  ->  Fun  F )
djufun.g  |-  ( ph  ->  Fun  G )
Assertion
Ref Expression
djufun  |-  ( ph  ->  Fun  ( F ⊔d  G ) )

Proof of Theorem djufun
StepHypRef Expression
1 djufun.f . . . 4  |-  ( ph  ->  Fun  F )
2 inlresf1 6939 . . . . 5  |-  (inl  |`  dom  F
) : dom  F -1-1-> ( dom  F dom  G )
3 df-f1 5123 . . . . . 6  |-  ( (inl  |`  dom  F ) : dom  F -1-1-> ( dom 
F dom  G )  <->  ( (inl  |` 
dom  F ) : dom  F --> ( dom 
F dom  G )  /\  Fun  `' (inl  |`  dom  F
) ) )
43simprbi 273 . . . . 5  |-  ( (inl  |`  dom  F ) : dom  F -1-1-> ( dom 
F dom  G )  ->  Fun  `' (inl  |`  dom  F
) )
52, 4mp1i 10 . . . 4  |-  ( ph  ->  Fun  `' (inl  |`  dom  F
) )
6 funco 5158 . . . 4  |-  ( ( Fun  F  /\  Fun  `' (inl  |`  dom  F ) )  ->  Fun  ( F  o.  `' (inl  |`  dom  F
) ) )
71, 5, 6syl2anc 408 . . 3  |-  ( ph  ->  Fun  ( F  o.  `' (inl  |`  dom  F
) ) )
8 djufun.g . . . 4  |-  ( ph  ->  Fun  G )
9 inrresf1 6940 . . . . 5  |-  (inr  |`  dom  G
) : dom  G -1-1-> ( dom  F dom  G )
10 df-f1 5123 . . . . . 6  |-  ( (inr  |`  dom  G ) : dom  G -1-1-> ( dom 
F dom  G )  <->  ( (inr  |` 
dom  G ) : dom  G --> ( dom 
F dom  G )  /\  Fun  `' (inr  |`  dom  G
) ) )
1110simprbi 273 . . . . 5  |-  ( (inr  |`  dom  G ) : dom  G -1-1-> ( dom 
F dom  G )  ->  Fun  `' (inr  |`  dom  G
) )
129, 11mp1i 10 . . . 4  |-  ( ph  ->  Fun  `' (inr  |`  dom  G
) )
13 funco 5158 . . . 4  |-  ( ( Fun  G  /\  Fun  `' (inr  |`  dom  G ) )  ->  Fun  ( G  o.  `' (inr  |`  dom  G
) ) )
148, 12, 13syl2anc 408 . . 3  |-  ( ph  ->  Fun  ( G  o.  `' (inr  |`  dom  G
) ) )
15 dmcoss 4803 . . . . . . 7  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) ) 
C_  dom  `' (inl  |` 
dom  F )
16 df-rn 4545 . . . . . . 7  |-  ran  (inl  |` 
dom  F )  =  dom  `' (inl  |`  dom  F
)
1715, 16sseqtrri 3127 . . . . . 6  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) ) 
C_  ran  (inl  |`  dom  F
)
18 dmcoss 4803 . . . . . . 7  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) ) 
C_  dom  `' (inr  |` 
dom  G )
19 df-rn 4545 . . . . . . 7  |-  ran  (inr  |` 
dom  G )  =  dom  `' (inr  |`  dom  G
)
2018, 19sseqtrri 3127 . . . . . 6  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) ) 
C_  ran  (inr  |`  dom  G
)
21 ss2in 3299 . . . . . 6  |-  ( ( dom  ( F  o.  `' (inl  |`  dom  F
) )  C_  ran  (inl  |`  dom  F )  /\  dom  ( G  o.  `' (inr  |`  dom  G
) )  C_  ran  (inr  |`  dom  G ) )  ->  ( dom  ( F  o.  `' (inl  |`  dom  F ) )  i^i  dom  ( G  o.  `' (inr  |` 
dom  G ) ) )  C_  ( ran  (inl  |`  dom  F )  i^i  ran  (inr  |`  dom  G
) ) )
2217, 20, 21mp2an 422 . . . . 5  |-  ( dom  ( F  o.  `' (inl  |`  dom  F ) )  i^i  dom  ( G  o.  `' (inr  |` 
dom  G ) ) )  C_  ( ran  (inl  |`  dom  F )  i^i  ran  (inr  |`  dom  G
) )
23 djuinr 6941 . . . . . 6  |-  ( ran  (inl  |`  dom  F )  i^i  ran  (inr  |`  dom  G
) )  =  (/)
2423a1i 9 . . . . 5  |-  ( ph  ->  ( ran  (inl  |`  dom  F
)  i^i  ran  (inr  |`  dom  G
) )  =  (/) )
2522, 24sseqtrid 3142 . . . 4  |-  ( ph  ->  ( dom  ( F  o.  `' (inl  |`  dom  F
) )  i^i  dom  ( G  o.  `' (inr  |`  dom  G ) ) )  C_  (/) )
26 ss0 3398 . . . 4  |-  ( ( dom  ( F  o.  `' (inl  |`  dom  F
) )  i^i  dom  ( G  o.  `' (inr  |`  dom  G ) ) )  C_  (/)  ->  ( dom  ( F  o.  `' (inl  |`  dom  F ) )  i^i  dom  ( G  o.  `' (inr  |` 
dom  G ) ) )  =  (/) )
2725, 26syl 14 . . 3  |-  ( ph  ->  ( dom  ( F  o.  `' (inl  |`  dom  F
) )  i^i  dom  ( G  o.  `' (inr  |`  dom  G ) ) )  =  (/) )
28 funun 5162 . . 3  |-  ( ( ( Fun  ( F  o.  `' (inl  |`  dom  F
) )  /\  Fun  ( G  o.  `' (inr  |`  dom  G ) ) )  /\  ( dom  ( F  o.  `' (inl  |`  dom  F ) )  i^i  dom  ( G  o.  `' (inr  |` 
dom  G ) ) )  =  (/) )  ->  Fun  ( ( F  o.  `' (inl  |`  dom  F
) )  u.  ( G  o.  `' (inr  |` 
dom  G ) ) ) )
297, 14, 27, 28syl21anc 1215 . 2  |-  ( ph  ->  Fun  ( ( F  o.  `' (inl  |`  dom  F
) )  u.  ( G  o.  `' (inr  |` 
dom  G ) ) ) )
30 df-djud 6981 . . 3  |-  ( F ⊔d  G )  =  ( ( F  o.  `' (inl  |`  dom  F ) )  u.  ( G  o.  `' (inr  |`  dom  G
) ) )
3130funeqi 5139 . 2  |-  ( Fun  ( F ⊔d  G )  <->  Fun  ( ( F  o.  `' (inl  |`  dom  F
) )  u.  ( G  o.  `' (inr  |` 
dom  G ) ) ) )
3229, 31sylibr 133 1  |-  ( ph  ->  Fun  ( F ⊔d  G ) )
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   -->wf 5114   -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-fal 1337  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-ne 2307  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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