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Theorem caseinl 7121
Description: Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
Hypotheses
Ref Expression
caseinl.f  |-  ( ph  ->  F  Fn  B )
caseinl.g  |-  ( ph  ->  Fun  G )
caseinl.a  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
caseinl  |-  ( ph  ->  (case ( F ,  G ) `  (inl `  A ) )  =  ( F `  A
) )

Proof of Theorem caseinl
StepHypRef Expression
1 df-case 7114 . . . 4  |- case ( F ,  G )  =  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
)
21fveq1i 5535 . . 3  |-  (case ( F ,  G ) `
 (inl `  A
) )  =  ( ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
) `  (inl `  A
) )
3 caseinl.f . . . . . . 7  |-  ( ph  ->  F  Fn  B )
4 fnfun 5332 . . . . . . 7  |-  ( F  Fn  B  ->  Fun  F )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  Fun  F )
6 djulf1o 7088 . . . . . . . 8  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
7 f1ocnv 5493 . . . . . . . 8  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  ->  `'inl : ( { (/) }  X.  _V ) -1-1-onto-> _V )
86, 7ax-mp 5 . . . . . . 7  |-  `'inl :
( { (/) }  X.  _V ) -1-1-onto-> _V
9 f1ofun 5482 . . . . . . 7  |-  ( `'inl
: ( { (/) }  X.  _V ) -1-1-onto-> _V  ->  Fun  `'inl )
108, 9ax-mp 5 . . . . . 6  |-  Fun  `'inl
11 funco 5275 . . . . . 6  |-  ( ( Fun  F  /\  Fun  `'inl )  ->  Fun  ( F  o.  `'inl ) )
125, 10, 11sylancl 413 . . . . 5  |-  ( ph  ->  Fun  ( F  o.  `'inl ) )
13 dmco 5155 . . . . . 6  |-  dom  ( F  o.  `'inl )  =  ( `' `'inl " dom  F )
14 df-inl 7077 . . . . . . . . . . 11  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
1514funmpt2 5274 . . . . . . . . . 10  |-  Fun inl
16 funrel 5252 . . . . . . . . . 10  |-  ( Fun inl  ->  Rel inl )
1715, 16ax-mp 5 . . . . . . . . 9  |-  Rel inl
18 dfrel2 5097 . . . . . . . . 9  |-  ( Rel inl  <->  `' `'inl  = inl )
1917, 18mpbi 145 . . . . . . . 8  |-  `' `'inl  = inl
2019a1i 9 . . . . . . 7  |-  ( ph  ->  `' `'inl  = inl )
21 fndm 5334 . . . . . . . 8  |-  ( F  Fn  B  ->  dom  F  =  B )
223, 21syl 14 . . . . . . 7  |-  ( ph  ->  dom  F  =  B )
2320, 22imaeq12d 4989 . . . . . 6  |-  ( ph  ->  ( `' `'inl " dom  F )  =  (inl " B ) )
2413, 23eqtrid 2234 . . . . 5  |-  ( ph  ->  dom  ( F  o.  `'inl )  =  (inl " B ) )
25 df-fn 5238 . . . . 5  |-  ( ( F  o.  `'inl )  Fn  (inl " B )  <-> 
( Fun  ( F  o.  `'inl )  /\  dom  ( F  o.  `'inl )  =  (inl " B ) ) )
2612, 24, 25sylanbrc 417 . . . 4  |-  ( ph  ->  ( F  o.  `'inl )  Fn  (inl " B
) )
27 caseinl.g . . . . . 6  |-  ( ph  ->  Fun  G )
28 djurf1o 7089 . . . . . . . 8  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
29 f1ocnv 5493 . . . . . . . 8  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  ->  `'inr :
( { 1o }  X.  _V ) -1-1-onto-> _V )
3028, 29ax-mp 5 . . . . . . 7  |-  `'inr :
( { 1o }  X.  _V ) -1-1-onto-> _V
31 f1ofun 5482 . . . . . . 7  |-  ( `'inr
: ( { 1o }  X.  _V ) -1-1-onto-> _V  ->  Fun  `'inr )
3230, 31ax-mp 5 . . . . . 6  |-  Fun  `'inr
33 funco 5275 . . . . . 6  |-  ( ( Fun  G  /\  Fun  `'inr )  ->  Fun  ( G  o.  `'inr ) )
3427, 32, 33sylancl 413 . . . . 5  |-  ( ph  ->  Fun  ( G  o.  `'inr ) )
35 dmco 5155 . . . . . . 7  |-  dom  ( G  o.  `'inr )  =  ( `' `'inr " dom  G )
36 imacnvcnv 5111 . . . . . . 7  |-  ( `' `'inr " dom  G )  =  (inr " dom  G )
3735, 36eqtri 2210 . . . . . 6  |-  dom  ( G  o.  `'inr )  =  (inr " dom  G
)
3837a1i 9 . . . . 5  |-  ( ph  ->  dom  ( G  o.  `'inr )  =  (inr " dom  G ) )
39 df-fn 5238 . . . . 5  |-  ( ( G  o.  `'inr )  Fn  (inr " dom  G
)  <->  ( Fun  ( G  o.  `'inr )  /\  dom  ( G  o.  `'inr )  =  (inr " dom  G ) ) )
4034, 38, 39sylanbrc 417 . . . 4  |-  ( ph  ->  ( G  o.  `'inr )  Fn  (inr " dom  G ) )
41 djuin 7094 . . . . 5  |-  ( (inl " B )  i^i  (inr " dom  G ) )  =  (/)
4241a1i 9 . . . 4  |-  ( ph  ->  ( (inl " B
)  i^i  (inr " dom  G ) )  =  (/) )
43 caseinl.a . . . . . . . 8  |-  ( ph  ->  A  e.  B )
4443elexd 2765 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
45 f1odm 5484 . . . . . . . 8  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  ->  dom inl  =  _V )
466, 45ax-mp 5 . . . . . . 7  |-  dom inl  =  _V
4744, 46eleqtrrdi 2283 . . . . . 6  |-  ( ph  ->  A  e.  dom inl )
4847, 15jctil 312 . . . . 5  |-  ( ph  ->  ( Fun inl  /\  A  e. 
dom inl ) )
49 funfvima 5769 . . . . 5  |-  ( ( Fun inl  /\  A  e.  dom inl )  ->  ( A  e.  B  ->  (inl `  A )  e.  (inl " B ) ) )
5048, 43, 49sylc 62 . . . 4  |-  ( ph  ->  (inl `  A )  e.  (inl " B ) )
51 fvun1 5603 . . . 4  |-  ( ( ( F  o.  `'inl )  Fn  (inl " B
)  /\  ( G  o.  `'inr )  Fn  (inr " dom  G )  /\  ( ( (inl " B )  i^i  (inr " dom  G ) )  =  (/)  /\  (inl `  A )  e.  (inl " B ) ) )  ->  ( ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) `  (inl `  A ) )  =  ( ( F  o.  `'inl ) `  (inl `  A ) ) )
5226, 40, 42, 50, 51syl112anc 1253 . . 3  |-  ( ph  ->  ( ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) `  (inl `  A ) )  =  ( ( F  o.  `'inl ) `  (inl `  A ) ) )
532, 52eqtrid 2234 . 2  |-  ( ph  ->  (case ( F ,  G ) `  (inl `  A ) )  =  ( ( F  o.  `'inl ) `  (inl `  A ) ) )
54 f1ofn 5481 . . . 4  |-  ( `'inl
: ( { (/) }  X.  _V ) -1-1-onto-> _V  ->  `'inl 
Fn  ( { (/) }  X.  _V ) )
558, 54ax-mp 5 . . 3  |-  `'inl  Fn  ( { (/) }  X.  _V )
56 f1of 5480 . . . . . 6  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  -> inl : _V --> ( {
(/) }  X.  _V )
)
576, 56ax-mp 5 . . . . 5  |- inl : _V --> ( { (/) }  X.  _V )
5857a1i 9 . . . 4  |-  ( ph  -> inl : _V --> ( {
(/) }  X.  _V )
)
5958, 44ffvelcdmd 5673 . . 3  |-  ( ph  ->  (inl `  A )  e.  ( { (/) }  X.  _V ) )
60 fvco2 5606 . . 3  |-  ( ( `'inl  Fn  ( { (/) }  X.  _V )  /\  (inl `  A )  e.  ( { (/) }  X.  _V ) )  ->  (
( F  o.  `'inl ) `  (inl `  A
) )  =  ( F `  ( `'inl `  (inl `  A )
) ) )
6155, 59, 60sylancr 414 . 2  |-  ( ph  ->  ( ( F  o.  `'inl ) `  (inl `  A ) )  =  ( F `  ( `'inl `  (inl `  A
) ) ) )
62 f1ocnvfv1 5799 . . . 4  |-  ( (inl : _V -1-1-onto-> ( { (/) }  X.  _V )  /\  A  e. 
_V )  ->  ( `'inl `  (inl `  A
) )  =  A )
636, 44, 62sylancr 414 . . 3  |-  ( ph  ->  ( `'inl `  (inl `  A ) )  =  A )
6463fveq2d 5538 . 2  |-  ( ph  ->  ( F `  ( `'inl `  (inl `  A
) ) )  =  ( F `  A
) )
6553, 61, 643eqtrd 2226 1  |-  ( ph  ->  (case ( F ,  G ) `  (inl `  A ) )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752    u. cun 3142    i^i cin 3143   (/)c0 3437   {csn 3607   <.cop 3610    X. cxp 4642   `'ccnv 4643   dom cdm 4644   "cima 4647    o. ccom 4648   Rel wrel 4649   Fun wfun 5229    Fn wfn 5230   -->wf 5231   -1-1-onto->wf1o 5234   ` cfv 5235   1oc1o 6435  inlcinl 7075  inrcinr 7076  casecdjucase 7113
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6166  df-2nd 6167  df-1o 6442  df-inl 7077  df-inr 7078  df-case 7114
This theorem is referenced by:  omp1eomlem  7124  ctm  7139
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