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Theorem casefun 6776
Description: The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
casefun.f  |-  ( ph  ->  Fun  F )
casefun.g  |-  ( ph  ->  Fun  G )
Assertion
Ref Expression
casefun  |-  ( ph  ->  Fun case ( F ,  G ) )

Proof of Theorem casefun
StepHypRef Expression
1 casefun.f . . . 4  |-  ( ph  ->  Fun  F )
2 djulf1o 6750 . . . . . 6  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
3 f1of1 5252 . . . . . 6  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  -> inl : _V -1-1-> ( {
(/) }  X.  _V )
)
42, 3ax-mp 7 . . . . 5  |- inl : _V -1-1-> ( { (/) }  X.  _V )
5 df-f1 5020 . . . . . 6  |-  (inl : _V
-1-1-> ( { (/) }  X.  _V )  <->  (inl : _V --> ( {
(/) }  X.  _V )  /\  Fun  `'inl ) )
65simprbi 269 . . . . 5  |-  (inl : _V
-1-1-> ( { (/) }  X.  _V )  ->  Fun  `'inl )
74, 6mp1i 10 . . . 4  |-  ( ph  ->  Fun  `'inl )
8 funco 5054 . . . 4  |-  ( ( Fun  F  /\  Fun  `'inl )  ->  Fun  ( F  o.  `'inl ) )
91, 7, 8syl2anc 403 . . 3  |-  ( ph  ->  Fun  ( F  o.  `'inl ) )
10 casefun.g . . . 4  |-  ( ph  ->  Fun  G )
11 djurf1o 6751 . . . . . 6  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
12 f1of1 5252 . . . . . 6  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  -> inr : _V -1-1-> ( { 1o }  X.  _V ) )
1311, 12ax-mp 7 . . . . 5  |- inr : _V -1-1-> ( { 1o }  X.  _V )
14 df-f1 5020 . . . . . 6  |-  (inr : _V
-1-1-> ( { 1o }  X.  _V )  <->  (inr : _V
--> ( { 1o }  X.  _V )  /\  Fun  `'inr ) )
1514simprbi 269 . . . . 5  |-  (inr : _V
-1-1-> ( { 1o }  X.  _V )  ->  Fun  `'inr )
1613, 15mp1i 10 . . . 4  |-  ( ph  ->  Fun  `'inr )
17 funco 5054 . . . 4  |-  ( ( Fun  G  /\  Fun  `'inr )  ->  Fun  ( G  o.  `'inr ) )
1810, 16, 17syl2anc 403 . . 3  |-  ( ph  ->  Fun  ( G  o.  `'inr ) )
19 dmcoss 4702 . . . . . . 7  |-  dom  ( F  o.  `'inl )  C_ 
dom  `'inl
20 df-rn 4449 . . . . . . 7  |-  ran inl  =  dom  `'inl
2119, 20sseqtr4i 3059 . . . . . 6  |-  dom  ( F  o.  `'inl )  C_ 
ran inl
22 dmcoss 4702 . . . . . . 7  |-  dom  ( G  o.  `'inr )  C_ 
dom  `'inr
23 df-rn 4449 . . . . . . 7  |-  ran inr  =  dom  `'inr
2422, 23sseqtr4i 3059 . . . . . 6  |-  dom  ( G  o.  `'inr )  C_ 
ran inr
25 ss2in 3227 . . . . . 6  |-  ( ( dom  ( F  o.  `'inl )  C_  ran inl  /\  dom  ( G  o.  `'inr )  C_  ran inr )  ->  ( dom  ( F  o.  `'inl )  i^i  dom  ( G  o.  `'inr )
)  C_  ( ran inl  i^i 
ran inr ) )
2621, 24, 25mp2an 417 . . . . 5  |-  ( dom  ( F  o.  `'inl )  i^i  dom  ( G  o.  `'inr ) )  C_  ( ran inl  i^i  ran inr )
27 rnresv 4890 . . . . . . . . 9  |-  ran  (inl  |` 
_V )  =  ran inl
2827eqcomi 2092 . . . . . . . 8  |-  ran inl  =  ran  (inl  |`  _V )
29 rnresv 4890 . . . . . . . . 9  |-  ran  (inr  |` 
_V )  =  ran inr
3029eqcomi 2092 . . . . . . . 8  |-  ran inr  =  ran  (inr  |`  _V )
3128, 30ineq12i 3199 . . . . . . 7  |-  ( ran inl  i^i  ran inr )  =  ( ran  (inl  |`  _V )  i^i  ran  (inr  |`  _V )
)
32 djuinr 6755 . . . . . . 7  |-  ( ran  (inl  |`  _V )  i^i 
ran  (inr  |`  _V )
)  =  (/)
3331, 32eqtri 2108 . . . . . 6  |-  ( ran inl  i^i  ran inr )  =  (/)
3433a1i 9 . . . . 5  |-  ( ph  ->  ( ran inl  i^i  ran inr )  =  (/) )
3526, 34syl5sseq 3074 . . . 4  |-  ( ph  ->  ( dom  ( F  o.  `'inl )  i^i 
dom  ( G  o.  `'inr ) )  C_  (/) )
36 ss0 3323 . . . 4  |-  ( ( dom  ( F  o.  `'inl )  i^i  dom  ( G  o.  `'inr )
)  C_  (/)  ->  ( dom  ( F  o.  `'inl )  i^i  dom  ( G  o.  `'inr ) )  =  (/) )
3735, 36syl 14 . . 3  |-  ( ph  ->  ( dom  ( F  o.  `'inl )  i^i 
dom  ( G  o.  `'inr ) )  =  (/) )
38 funun 5058 . . 3  |-  ( ( ( Fun  ( F  o.  `'inl )  /\  Fun  ( G  o.  `'inr ) )  /\  ( dom  ( F  o.  `'inl )  i^i  dom  ( G  o.  `'inr ) )  =  (/) )  ->  Fun  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) )
399, 18, 37, 38syl21anc 1173 . 2  |-  ( ph  ->  Fun  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) )
40 df-case 6775 . . 3  |- case ( F ,  G )  =  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
)
4140funeqi 5036 . 2  |-  ( Fun case
( F ,  G
)  <->  Fun  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) )
4239, 41sylibr 132 1  |-  ( ph  ->  Fun case ( F ,  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   _Vcvv 2619    u. cun 2997    i^i cin 2998    C_ wss 2999   (/)c0 3286   {csn 3446    X. cxp 4436   `'ccnv 4437   dom cdm 4438   ran crn 4439    |` cres 4440    o. ccom 4442   Fun wfun 5009   -->wf 5011   -1-1->wf1 5012   -1-1-onto->wf1o 5014   1oc1o 6174  inlcinl 6737  inrcinr 6738  casecdjucase 6774
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-inl 6739  df-inr 6740  df-case 6775
This theorem is referenced by:  casef  6779
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