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Theorem casef 6941
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 5245 . . . . 5  |-  ( F : A --> X  ->  Fun  F )
31, 2syl 14 . . . 4  |-  ( ph  ->  Fun  F )
4 casef.g . . . . 5  |-  ( ph  ->  G : B --> X )
5 ffun 5245 . . . . 5  |-  ( G : B --> X  ->  Fun  G )
64, 5syl 14 . . . 4  |-  ( ph  ->  Fun  G )
73, 6casefun 6938 . . 3  |-  ( ph  ->  Fun case ( F ,  G ) )
8 caserel 6940 . . . 4  |- case ( F ,  G )  C_  ( ( dom  F dom 
G )  X.  ( ran  F  u.  ran  G
) )
9 ssid 3087 . . . . 5  |-  ( dom 
F dom  G )  C_  ( dom  F dom  G )
10 frn 5251 . . . . . . 7  |-  ( F : A --> X  ->  ran  F  C_  X )
111, 10syl 14 . . . . . 6  |-  ( ph  ->  ran  F  C_  X
)
12 frn 5251 . . . . . . 7  |-  ( G : B --> X  ->  ran  G  C_  X )
134, 12syl 14 . . . . . 6  |-  ( ph  ->  ran  G  C_  X
)
1411, 13unssd 3222 . . . . 5  |-  ( ph  ->  ( ran  F  u.  ran  G )  C_  X
)
15 xpss12 4616 . . . . 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 3078 . . 3  |-  ( ph  -> case ( F ,  G
)  C_  ( ( dom  F dom  G )  X.  X ) )
18 funssxp 5262 . . . 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 6939 . . . 4  |-  dom case ( F ,  G )  =  ( dom  F dom 
G )
22 fdm 5248 . . . . . 6  |-  ( F : A --> X  ->  dom  F  =  A )
231, 22syl 14 . . . . 5  |-  ( ph  ->  dom  F  =  A )
24 fdm 5248 . . . . . 6  |-  ( G : B --> X  ->  dom  G  =  B )
254, 24syl 14 . . . . 5  |-  ( ph  ->  dom  G  =  B )
26 djueq12 6892 . . . . 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 2162 . . 3  |-  ( ph  ->  dom case ( F ,  G )  =  ( A B ) )
2928feq2d 5230 . 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 1316    u. cun 3039    C_ wss 3041    X. cxp 4507   dom cdm 4509   ran crn 4510   Fun wfun 5087   -->wf 5089   ⊔ cdju 6890  casecdjucase 6936
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-fal 1322  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-ne 2286  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-ima 4522  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-case 6937
This theorem is referenced by:  casef1  6943  omp1eomlem  6947  ctm  6962
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