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Theorem djufun 6784
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 6753 . . . . 5  |-  (inl  |`  dom  F
) : dom  F -1-1-> ( dom  F dom  G )
3 df-f1 5020 . . . . . 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 269 . . . . 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 5054 . . . 4  |-  ( ( Fun  F  /\  Fun  `' (inl  |`  dom  F ) )  ->  Fun  ( F  o.  `' (inl  |`  dom  F
) ) )
71, 5, 6syl2anc 403 . . 3  |-  ( ph  ->  Fun  ( F  o.  `' (inl  |`  dom  F
) ) )
8 djufun.g . . . 4  |-  ( ph  ->  Fun  G )
9 inrresf1 6754 . . . . 5  |-  (inr  |`  dom  G
) : dom  G -1-1-> ( dom  F dom  G )
10 df-f1 5020 . . . . . 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 269 . . . . 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 5054 . . . 4  |-  ( ( Fun  G  /\  Fun  `' (inr  |`  dom  G ) )  ->  Fun  ( G  o.  `' (inr  |`  dom  G
) ) )
148, 12, 13syl2anc 403 . . 3  |-  ( ph  ->  Fun  ( G  o.  `' (inr  |`  dom  G
) ) )
15 dmcoss 4702 . . . . . . 7  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) ) 
C_  dom  `' (inl  |` 
dom  F )
16 df-rn 4449 . . . . . . 7  |-  ran  (inl  |` 
dom  F )  =  dom  `' (inl  |`  dom  F
)
1715, 16sseqtr4i 3059 . . . . . 6  |-  dom  ( F  o.  `' (inl  |` 
dom  F ) ) 
C_  ran  (inl  |`  dom  F
)
18 dmcoss 4702 . . . . . . 7  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) ) 
C_  dom  `' (inr  |` 
dom  G )
19 df-rn 4449 . . . . . . 7  |-  ran  (inr  |` 
dom  G )  =  dom  `' (inr  |`  dom  G
)
2018, 19sseqtr4i 3059 . . . . . 6  |-  dom  ( G  o.  `' (inr  |` 
dom  G ) ) 
C_  ran  (inr  |`  dom  G
)
21 ss2in 3227 . . . . . 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 417 . . . . 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 6755 . . . . . 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, 24syl5sseq 3074 . . . 4  |-  ( ph  ->  ( dom  ( F  o.  `' (inl  |`  dom  F
) )  i^i  dom  ( G  o.  `' (inr  |`  dom  G ) ) )  C_  (/) )
26 ss0 3323 . . . 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 5058 . . 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 1173 . 2  |-  ( ph  ->  Fun  ( ( F  o.  `' (inl  |`  dom  F
) )  u.  ( G  o.  `' (inr  |` 
dom  G ) ) ) )
30 df-djud 6783 . . 3  |-  ( F ⊔d  G )  =  ( ( F  o.  `' (inl  |`  dom  F ) )  u.  ( G  o.  `' (inr  |`  dom  G
) ) )
3130funeqi 5036 . 2  |-  ( Fun  ( F ⊔d  G )  <->  Fun  ( ( F  o.  `' (inl  |`  dom  F
) )  u.  ( G  o.  `' (inr  |` 
dom  G ) ) ) )
3229, 31sylibr 132 1  |-  ( ph  ->  Fun  ( F ⊔d  G ) )
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   -->wf 5011   -1-1->wf1 5012   ⊔ cdju 6730  inlcinl 6737  inrcinr 6738   ⊔d cdjud 6782
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-fal 1295  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-ne 2256  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 6731  df-inl 6739  df-inr 6740  df-djud 6783
This theorem is referenced by: (None)
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