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Theorem caseinl 7080
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 7073 . . . 4  |- case ( F ,  G )  =  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
)
21fveq1i 5508 . . 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 5305 . . . . . . 7  |-  ( F  Fn  B  ->  Fun  F )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  Fun  F )
6 djulf1o 7047 . . . . . . . 8  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
7 f1ocnv 5466 . . . . . . . 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 5455 . . . . . . 7  |-  ( `'inl
: ( { (/) }  X.  _V ) -1-1-onto-> _V  ->  Fun  `'inl )
108, 9ax-mp 5 . . . . . 6  |-  Fun  `'inl
11 funco 5248 . . . . . 6  |-  ( ( Fun  F  /\  Fun  `'inl )  ->  Fun  ( F  o.  `'inl ) )
125, 10, 11sylancl 413 . . . . 5  |-  ( ph  ->  Fun  ( F  o.  `'inl ) )
13 dmco 5129 . . . . . 6  |-  dom  ( F  o.  `'inl )  =  ( `' `'inl " dom  F )
14 df-inl 7036 . . . . . . . . . . 11  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
1514funmpt2 5247 . . . . . . . . . 10  |-  Fun inl
16 funrel 5225 . . . . . . . . . 10  |-  ( Fun inl  ->  Rel inl )
1715, 16ax-mp 5 . . . . . . . . 9  |-  Rel inl
18 dfrel2 5071 . . . . . . . . 9  |-  ( Rel inl  <->  `' `'inl  = inl )
1917, 18mpbi 145 . . . . . . . 8  |-  `' `'inl  = inl
2019a1i 9 . . . . . . 7  |-  ( ph  ->  `' `'inl  = inl )
21 fndm 5307 . . . . . . . 8  |-  ( F  Fn  B  ->  dom  F  =  B )
223, 21syl 14 . . . . . . 7  |-  ( ph  ->  dom  F  =  B )
2320, 22imaeq12d 4964 . . . . . 6  |-  ( ph  ->  ( `' `'inl " dom  F )  =  (inl " B ) )
2413, 23eqtrid 2220 . . . . 5  |-  ( ph  ->  dom  ( F  o.  `'inl )  =  (inl " B ) )
25 df-fn 5211 . . . . 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 7048 . . . . . . . 8  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
29 f1ocnv 5466 . . . . . . . 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 5455 . . . . . . 7  |-  ( `'inr
: ( { 1o }  X.  _V ) -1-1-onto-> _V  ->  Fun  `'inr )
3230, 31ax-mp 5 . . . . . 6  |-  Fun  `'inr
33 funco 5248 . . . . . 6  |-  ( ( Fun  G  /\  Fun  `'inr )  ->  Fun  ( G  o.  `'inr ) )
3427, 32, 33sylancl 413 . . . . 5  |-  ( ph  ->  Fun  ( G  o.  `'inr ) )
35 dmco 5129 . . . . . . 7  |-  dom  ( G  o.  `'inr )  =  ( `' `'inr " dom  G )
36 imacnvcnv 5085 . . . . . . 7  |-  ( `' `'inr " dom  G )  =  (inr " dom  G )
3735, 36eqtri 2196 . . . . . 6  |-  dom  ( G  o.  `'inr )  =  (inr " dom  G
)
3837a1i 9 . . . . 5  |-  ( ph  ->  dom  ( G  o.  `'inr )  =  (inr " dom  G ) )
39 df-fn 5211 . . . . 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 7053 . . . . 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 2748 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
45 f1odm 5457 . . . . . . . 8  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  ->  dom inl  =  _V )
466, 45ax-mp 5 . . . . . . 7  |-  dom inl  =  _V
4744, 46eleqtrrdi 2269 . . . . . 6  |-  ( ph  ->  A  e.  dom inl )
4847, 15jctil 312 . . . . 5  |-  ( ph  ->  ( Fun inl  /\  A  e. 
dom inl ) )
49 funfvima 5739 . . . . 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 5574 . . . 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 1242 . . 3  |-  ( ph  ->  ( ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) `  (inl `  A ) )  =  ( ( F  o.  `'inl ) `  (inl `  A ) ) )
532, 52eqtrid 2220 . 2  |-  ( ph  ->  (case ( F ,  G ) `  (inl `  A ) )  =  ( ( F  o.  `'inl ) `  (inl `  A ) ) )
54 f1ofn 5454 . . . 4  |-  ( `'inl
: ( { (/) }  X.  _V ) -1-1-onto-> _V  ->  `'inl 
Fn  ( { (/) }  X.  _V ) )
558, 54ax-mp 5 . . 3  |-  `'inl  Fn  ( { (/) }  X.  _V )
56 f1of 5453 . . . . . 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 5644 . . 3  |-  ( ph  ->  (inl `  A )  e.  ( { (/) }  X.  _V ) )
60 fvco2 5577 . . 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 5768 . . . 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 5511 . 2  |-  ( ph  ->  ( F `  ( `'inl `  (inl `  A
) ) )  =  ( F `  A
) )
6553, 61, 643eqtrd 2212 1  |-  ( ph  ->  (case ( F ,  G ) `  (inl `  A ) )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2146   _Vcvv 2735    u. cun 3125    i^i cin 3126   (/)c0 3420   {csn 3589   <.cop 3592    X. cxp 4618   `'ccnv 4619   dom cdm 4620   "cima 4623    o. ccom 4624   Rel wrel 4625   Fun wfun 5202    Fn wfn 5203   -->wf 5204   -1-1-onto->wf1o 5207   ` cfv 5208   1oc1o 6400  inlcinl 7034  inrcinr 7035  casecdjucase 7072
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-inl 7036  df-inr 7037  df-case 7073
This theorem is referenced by:  omp1eomlem  7083  ctm  7098
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