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Theorem casefun 6963
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 6936 . . . . . 6  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
3 f1of1 5359 . . . . . 6  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  -> inl : _V -1-1-> ( {
(/) }  X.  _V )
)
42, 3ax-mp 5 . . . . 5  |- inl : _V -1-1-> ( { (/) }  X.  _V )
5 df-f1 5123 . . . . . 6  |-  (inl : _V
-1-1-> ( { (/) }  X.  _V )  <->  (inl : _V --> ( {
(/) }  X.  _V )  /\  Fun  `'inl ) )
65simprbi 273 . . . . 5  |-  (inl : _V
-1-1-> ( { (/) }  X.  _V )  ->  Fun  `'inl )
74, 6mp1i 10 . . . 4  |-  ( ph  ->  Fun  `'inl )
8 funco 5158 . . . 4  |-  ( ( Fun  F  /\  Fun  `'inl )  ->  Fun  ( F  o.  `'inl ) )
91, 7, 8syl2anc 408 . . 3  |-  ( ph  ->  Fun  ( F  o.  `'inl ) )
10 casefun.g . . . 4  |-  ( ph  ->  Fun  G )
11 djurf1o 6937 . . . . . 6  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
12 f1of1 5359 . . . . . 6  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  -> inr : _V -1-1-> ( { 1o }  X.  _V ) )
1311, 12ax-mp 5 . . . . 5  |- inr : _V -1-1-> ( { 1o }  X.  _V )
14 df-f1 5123 . . . . . 6  |-  (inr : _V
-1-1-> ( { 1o }  X.  _V )  <->  (inr : _V
--> ( { 1o }  X.  _V )  /\  Fun  `'inr ) )
1514simprbi 273 . . . . 5  |-  (inr : _V
-1-1-> ( { 1o }  X.  _V )  ->  Fun  `'inr )
1613, 15mp1i 10 . . . 4  |-  ( ph  ->  Fun  `'inr )
17 funco 5158 . . . 4  |-  ( ( Fun  G  /\  Fun  `'inr )  ->  Fun  ( G  o.  `'inr ) )
1810, 16, 17syl2anc 408 . . 3  |-  ( ph  ->  Fun  ( G  o.  `'inr ) )
19 dmcoss 4803 . . . . . . 7  |-  dom  ( F  o.  `'inl )  C_ 
dom  `'inl
20 df-rn 4545 . . . . . . 7  |-  ran inl  =  dom  `'inl
2119, 20sseqtrri 3127 . . . . . 6  |-  dom  ( F  o.  `'inl )  C_ 
ran inl
22 dmcoss 4803 . . . . . . 7  |-  dom  ( G  o.  `'inr )  C_ 
dom  `'inr
23 df-rn 4545 . . . . . . 7  |-  ran inr  =  dom  `'inr
2422, 23sseqtrri 3127 . . . . . 6  |-  dom  ( G  o.  `'inr )  C_ 
ran inr
25 ss2in 3299 . . . . . 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 422 . . . . 5  |-  ( dom  ( F  o.  `'inl )  i^i  dom  ( G  o.  `'inr ) )  C_  ( ran inl  i^i  ran inr )
27 rnresv 4993 . . . . . . . . 9  |-  ran  (inl  |` 
_V )  =  ran inl
2827eqcomi 2141 . . . . . . . 8  |-  ran inl  =  ran  (inl  |`  _V )
29 rnresv 4993 . . . . . . . . 9  |-  ran  (inr  |` 
_V )  =  ran inr
3029eqcomi 2141 . . . . . . . 8  |-  ran inr  =  ran  (inr  |`  _V )
3128, 30ineq12i 3270 . . . . . . 7  |-  ( ran inl  i^i  ran inr )  =  ( ran  (inl  |`  _V )  i^i  ran  (inr  |`  _V )
)
32 djuinr 6941 . . . . . . 7  |-  ( ran  (inl  |`  _V )  i^i 
ran  (inr  |`  _V )
)  =  (/)
3331, 32eqtri 2158 . . . . . 6  |-  ( ran inl  i^i  ran inr )  =  (/)
3433a1i 9 . . . . 5  |-  ( ph  ->  ( ran inl  i^i  ran inr )  =  (/) )
3526, 34sseqtrid 3142 . . . 4  |-  ( ph  ->  ( dom  ( F  o.  `'inl )  i^i 
dom  ( G  o.  `'inr ) )  C_  (/) )
36 ss0 3398 . . . 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 5162 . . 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 1215 . 2  |-  ( ph  ->  Fun  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) )
40 df-case 6962 . . 3  |- case ( F ,  G )  =  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
)
4140funeqi 5139 . 2  |-  ( Fun case
( F ,  G
)  <->  Fun  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) )
4239, 41sylibr 133 1  |-  ( ph  ->  Fun case ( F ,  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   _Vcvv 2681    u. cun 3064    i^i cin 3065    C_ wss 3066   (/)c0 3358   {csn 3522    X. cxp 4532   `'ccnv 4533   dom cdm 4534   ran crn 4535    |` cres 4536    o. ccom 4538   Fun wfun 5112   -->wf 5114   -1-1->wf1 5115   -1-1-onto->wf1o 5117   1oc1o 6299  inlcinl 6923  inrcinr 6924  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-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-inl 6925  df-inr 6926  df-case 6962
This theorem is referenced by:  casef  6966
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