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Theorem casef 6833
Description: The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
casef.f  |-  ( ph  ->  F : A --> X )
casef.g  |-  ( ph  ->  G : B --> X )
Assertion
Ref Expression
casef  |-  ( ph  -> case ( F ,  G
) : ( A B ) --> X )

Proof of Theorem casef
StepHypRef Expression
1 casef.f . . . . 5  |-  ( ph  ->  F : A --> X )
2 ffun 5177 . . . . 5  |-  ( F : A --> X  ->  Fun  F )
31, 2syl 14 . . . 4  |-  ( ph  ->  Fun  F )
4 casef.g . . . . 5  |-  ( ph  ->  G : B --> X )
5 ffun 5177 . . . . 5  |-  ( G : B --> X  ->  Fun  G )
64, 5syl 14 . . . 4  |-  ( ph  ->  Fun  G )
73, 6casefun 6830 . . 3  |-  ( ph  ->  Fun case ( F ,  G ) )
8 caserel 6832 . . . 4  |- case ( F ,  G )  C_  ( ( dom  F dom 
G )  X.  ( ran  F  u.  ran  G
) )
9 ssid 3045 . . . . 5  |-  ( dom 
F dom  G )  C_  ( dom  F dom  G )
10 frn 5182 . . . . . . 7  |-  ( F : A --> X  ->  ran  F  C_  X )
111, 10syl 14 . . . . . 6  |-  ( ph  ->  ran  F  C_  X
)
12 frn 5182 . . . . . . 7  |-  ( G : B --> X  ->  ran  G  C_  X )
134, 12syl 14 . . . . . 6  |-  ( ph  ->  ran  G  C_  X
)
1411, 13unssd 3177 . . . . 5  |-  ( ph  ->  ( ran  F  u.  ran  G )  C_  X
)
15 xpss12 4558 . . . . 5  |-  ( ( ( dom  F dom  G
)  C_  ( dom  F dom  G )  /\  ( ran  F  u.  ran  G
)  C_  X )  ->  ( ( dom  F dom 
G )  X.  ( ran  F  u.  ran  G
) )  C_  (
( dom  F dom  G
)  X.  X ) )
169, 14, 15sylancr 406 . . . 4  |-  ( ph  ->  ( ( dom  F dom 
G )  X.  ( ran  F  u.  ran  G
) )  C_  (
( dom  F dom  G
)  X.  X ) )
178, 16syl5ss 3037 . . 3  |-  ( ph  -> case ( F ,  G
)  C_  ( ( dom  F dom  G )  X.  X ) )
18 funssxp 5193 . . . 4  |-  ( ( Fun case ( F ,  G )  /\ case ( F ,  G )  C_  ( ( dom  F dom 
G )  X.  X
) )  <->  (case ( F ,  G ) : dom case ( F ,  G ) --> X  /\  dom case ( F ,  G
)  C_  ( dom  F dom  G ) ) )
1918simplbi 269 . . 3  |-  ( ( Fun case ( F ,  G )  /\ case ( F ,  G )  C_  ( ( dom  F dom 
G )  X.  X
) )  -> case ( F ,  G ) : dom case ( F ,  G ) --> X )
207, 17, 19syl2anc 404 . 2  |-  ( ph  -> case ( F ,  G
) : dom case ( F ,  G ) --> X )
21 casedm 6831 . . . 4  |-  dom case ( F ,  G )  =  ( dom  F dom 
G )
22 fdm 5179 . . . . . 6  |-  ( F : A --> X  ->  dom  F  =  A )
231, 22syl 14 . . . . 5  |-  ( ph  ->  dom  F  =  A )
24 fdm 5179 . . . . . 6  |-  ( G : B --> X  ->  dom  G  =  B )
254, 24syl 14 . . . . 5  |-  ( ph  ->  dom  G  =  B )
26 djueq12 6786 . . . . 5  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F dom  G )  =  ( A B )
)
2723, 25, 26syl2anc 404 . . . 4  |-  ( ph  ->  ( dom  F dom  G
)  =  ( A B ) )
2821, 27syl5eq 2133 . . 3  |-  ( ph  ->  dom case ( F ,  G )  =  ( A B ) )
2928feq2d 5163 . 2  |-  ( ph  ->  (case ( F ,  G ) : dom case ( F ,  G ) --> X  <-> case ( F ,  G
) : ( A B ) --> X ) )
3020, 29mpbid 146 1  |-  ( ph  -> case ( F ,  G
) : ( A B ) --> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    u. cun 2998    C_ wss 3000    X. cxp 4450   dom cdm 4452   ran crn 4453   Fun wfun 5022   -->wf 5024   ⊔ cdju 6784  casecdjucase 6828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-1st 5925  df-2nd 5926  df-1o 6195  df-dju 6785  df-inl 6793  df-inr 6794  df-case 6829
This theorem is referenced by:  casef1  6835  ctm  6845
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