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Theorem casef 6966
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 5270 . . . . 5  |-  ( F : A --> X  ->  Fun  F )
31, 2syl 14 . . . 4  |-  ( ph  ->  Fun  F )
4 casef.g . . . . 5  |-  ( ph  ->  G : B --> X )
5 ffun 5270 . . . . 5  |-  ( G : B --> X  ->  Fun  G )
64, 5syl 14 . . . 4  |-  ( ph  ->  Fun  G )
73, 6casefun 6963 . . 3  |-  ( ph  ->  Fun case ( F ,  G ) )
8 caserel 6965 . . . 4  |- case ( F ,  G )  C_  ( ( dom  F dom 
G )  X.  ( ran  F  u.  ran  G
) )
9 ssid 3112 . . . . 5  |-  ( dom 
F dom  G )  C_  ( dom  F dom  G )
10 frn 5276 . . . . . . 7  |-  ( F : A --> X  ->  ran  F  C_  X )
111, 10syl 14 . . . . . 6  |-  ( ph  ->  ran  F  C_  X
)
12 frn 5276 . . . . . . 7  |-  ( G : B --> X  ->  ran  G  C_  X )
134, 12syl 14 . . . . . 6  |-  ( ph  ->  ran  G  C_  X
)
1411, 13unssd 3247 . . . . 5  |-  ( ph  ->  ( ran  F  u.  ran  G )  C_  X
)
15 xpss12 4641 . . . . 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 410 . . . 4  |-  ( ph  ->  ( ( dom  F dom 
G )  X.  ( ran  F  u.  ran  G
) )  C_  (
( dom  F dom  G
)  X.  X ) )
178, 16sstrid 3103 . . 3  |-  ( ph  -> case ( F ,  G
)  C_  ( ( dom  F dom  G )  X.  X ) )
18 funssxp 5287 . . . 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 272 . . 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 408 . 2  |-  ( ph  -> case ( F ,  G
) : dom case ( F ,  G ) --> X )
21 casedm 6964 . . . 4  |-  dom case ( F ,  G )  =  ( dom  F dom 
G )
22 fdm 5273 . . . . . 6  |-  ( F : A --> X  ->  dom  F  =  A )
231, 22syl 14 . . . . 5  |-  ( ph  ->  dom  F  =  A )
24 fdm 5273 . . . . . 6  |-  ( G : B --> X  ->  dom  G  =  B )
254, 24syl 14 . . . . 5  |-  ( ph  ->  dom  G  =  B )
26 djueq12 6917 . . . . 5  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F dom  G )  =  ( A B )
)
2723, 25, 26syl2anc 408 . . . 4  |-  ( ph  ->  ( dom  F dom  G
)  =  ( A B ) )
2821, 27syl5eq 2182 . . 3  |-  ( ph  ->  dom case ( F ,  G )  =  ( A B ) )
2928feq2d 5255 . 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 1331    u. cun 3064    C_ wss 3066    X. cxp 4532   dom cdm 4534   ran crn 4535   Fun wfun 5112   -->wf 5114   ⊔ cdju 6915  casecdjucase 6961
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-ima 4547  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-case 6962
This theorem is referenced by:  casef1  6968  omp1eomlem  6972  ctm  6987
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