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Theorem casef 7101
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 5380 . . . . 5  |-  ( F : A --> X  ->  Fun  F )
31, 2syl 14 . . . 4  |-  ( ph  ->  Fun  F )
4 casef.g . . . . 5  |-  ( ph  ->  G : B --> X )
5 ffun 5380 . . . . 5  |-  ( G : B --> X  ->  Fun  G )
64, 5syl 14 . . . 4  |-  ( ph  ->  Fun  G )
73, 6casefun 7098 . . 3  |-  ( ph  ->  Fun case ( F ,  G ) )
8 caserel 7100 . . . 4  |- case ( F ,  G )  C_  ( ( dom  F dom 
G )  X.  ( ran  F  u.  ran  G
) )
9 ssid 3187 . . . . 5  |-  ( dom 
F dom  G )  C_  ( dom  F dom  G )
10 frn 5386 . . . . . . 7  |-  ( F : A --> X  ->  ran  F  C_  X )
111, 10syl 14 . . . . . 6  |-  ( ph  ->  ran  F  C_  X
)
12 frn 5386 . . . . . . 7  |-  ( G : B --> X  ->  ran  G  C_  X )
134, 12syl 14 . . . . . 6  |-  ( ph  ->  ran  G  C_  X
)
1411, 13unssd 3323 . . . . 5  |-  ( ph  ->  ( ran  F  u.  ran  G )  C_  X
)
15 xpss12 4745 . . . . 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 414 . . . 4  |-  ( ph  ->  ( ( dom  F dom 
G )  X.  ( ran  F  u.  ran  G
) )  C_  (
( dom  F dom  G
)  X.  X ) )
178, 16sstrid 3178 . . 3  |-  ( ph  -> case ( F ,  G
)  C_  ( ( dom  F dom  G )  X.  X ) )
18 funssxp 5397 . . . 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 274 . . 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 411 . 2  |-  ( ph  -> case ( F ,  G
) : dom case ( F ,  G ) --> X )
21 casedm 7099 . . . 4  |-  dom case ( F ,  G )  =  ( dom  F dom 
G )
22 fdm 5383 . . . . . 6  |-  ( F : A --> X  ->  dom  F  =  A )
231, 22syl 14 . . . . 5  |-  ( ph  ->  dom  F  =  A )
24 fdm 5383 . . . . . 6  |-  ( G : B --> X  ->  dom  G  =  B )
254, 24syl 14 . . . . 5  |-  ( ph  ->  dom  G  =  B )
26 djueq12 7052 . . . . 5  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F dom  G )  =  ( A B )
)
2723, 25, 26syl2anc 411 . . . 4  |-  ( ph  ->  ( dom  F dom  G
)  =  ( A B ) )
2821, 27eqtrid 2232 . . 3  |-  ( ph  ->  dom case ( F ,  G )  =  ( A B ) )
2928feq2d 5365 . 2  |-  ( ph  ->  (case ( F ,  G ) : dom case ( F ,  G ) --> X  <-> case ( F ,  G
) : ( A B ) --> X ) )
3020, 29mpbid 147 1  |-  ( ph  -> case ( F ,  G
) : ( A B ) --> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    u. cun 3139    C_ wss 3141    X. cxp 4636   dom cdm 4638   ran crn 4639   Fun wfun 5222   -->wf 5224   ⊔ cdju 7050  casecdjucase 7096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6155  df-2nd 6156  df-1o 6431  df-dju 7051  df-inl 7060  df-inr 7061  df-case 7097
This theorem is referenced by:  casef1  7103  omp1eomlem  7107  ctm  7122
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