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Theorem caseinl 6862
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 6855 . . . 4  |- case ( F ,  G )  =  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
)
21fveq1i 5341 . . 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 5145 . . . . . . 7  |-  ( F  Fn  B  ->  Fun  F )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  Fun  F )
6 djulf1o 6830 . . . . . . . 8  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
7 f1ocnv 5301 . . . . . . . 8  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  ->  `'inl : ( { (/) }  X.  _V ) -1-1-onto-> _V )
86, 7ax-mp 7 . . . . . . 7  |-  `'inl :
( { (/) }  X.  _V ) -1-1-onto-> _V
9 f1ofun 5290 . . . . . . 7  |-  ( `'inl
: ( { (/) }  X.  _V ) -1-1-onto-> _V  ->  Fun  `'inl )
108, 9ax-mp 7 . . . . . 6  |-  Fun  `'inl
11 funco 5088 . . . . . 6  |-  ( ( Fun  F  /\  Fun  `'inl )  ->  Fun  ( F  o.  `'inl ) )
125, 10, 11sylancl 405 . . . . 5  |-  ( ph  ->  Fun  ( F  o.  `'inl ) )
13 dmco 4973 . . . . . 6  |-  dom  ( F  o.  `'inl )  =  ( `' `'inl " dom  F )
14 df-inl 6819 . . . . . . . . . . 11  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
1514funmpt2 5087 . . . . . . . . . 10  |-  Fun inl
16 funrel 5066 . . . . . . . . . 10  |-  ( Fun inl  ->  Rel inl )
1715, 16ax-mp 7 . . . . . . . . 9  |-  Rel inl
18 dfrel2 4915 . . . . . . . . 9  |-  ( Rel inl  <->  `' `'inl  = inl )
1917, 18mpbi 144 . . . . . . . 8  |-  `' `'inl  = inl
2019a1i 9 . . . . . . 7  |-  ( ph  ->  `' `'inl  = inl )
21 fndm 5147 . . . . . . . 8  |-  ( F  Fn  B  ->  dom  F  =  B )
223, 21syl 14 . . . . . . 7  |-  ( ph  ->  dom  F  =  B )
2320, 22imaeq12d 4808 . . . . . 6  |-  ( ph  ->  ( `' `'inl " dom  F )  =  (inl " B ) )
2413, 23syl5eq 2139 . . . . 5  |-  ( ph  ->  dom  ( F  o.  `'inl )  =  (inl " B ) )
25 df-fn 5052 . . . . 5  |-  ( ( F  o.  `'inl )  Fn  (inl " B )  <-> 
( Fun  ( F  o.  `'inl )  /\  dom  ( F  o.  `'inl )  =  (inl " B ) ) )
2612, 24, 25sylanbrc 409 . . . 4  |-  ( ph  ->  ( F  o.  `'inl )  Fn  (inl " B
) )
27 caseinl.g . . . . . 6  |-  ( ph  ->  Fun  G )
28 djurf1o 6831 . . . . . . . 8  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
29 f1ocnv 5301 . . . . . . . 8  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  ->  `'inr :
( { 1o }  X.  _V ) -1-1-onto-> _V )
3028, 29ax-mp 7 . . . . . . 7  |-  `'inr :
( { 1o }  X.  _V ) -1-1-onto-> _V
31 f1ofun 5290 . . . . . . 7  |-  ( `'inr
: ( { 1o }  X.  _V ) -1-1-onto-> _V  ->  Fun  `'inr )
3230, 31ax-mp 7 . . . . . 6  |-  Fun  `'inr
33 funco 5088 . . . . . 6  |-  ( ( Fun  G  /\  Fun  `'inr )  ->  Fun  ( G  o.  `'inr ) )
3427, 32, 33sylancl 405 . . . . 5  |-  ( ph  ->  Fun  ( G  o.  `'inr ) )
35 dmco 4973 . . . . . . 7  |-  dom  ( G  o.  `'inr )  =  ( `' `'inr " dom  G )
36 imacnvcnv 4929 . . . . . . 7  |-  ( `' `'inr " dom  G )  =  (inr " dom  G )
3735, 36eqtri 2115 . . . . . 6  |-  dom  ( G  o.  `'inr )  =  (inr " dom  G
)
3837a1i 9 . . . . 5  |-  ( ph  ->  dom  ( G  o.  `'inr )  =  (inr " dom  G ) )
39 df-fn 5052 . . . . 5  |-  ( ( G  o.  `'inr )  Fn  (inr " dom  G
)  <->  ( Fun  ( G  o.  `'inr )  /\  dom  ( G  o.  `'inr )  =  (inr " dom  G ) ) )
4034, 38, 39sylanbrc 409 . . . 4  |-  ( ph  ->  ( G  o.  `'inr )  Fn  (inr " dom  G ) )
41 djuin 6836 . . . . 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 2646 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
45 f1odm 5292 . . . . . . . 8  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  ->  dom inl  =  _V )
466, 45ax-mp 7 . . . . . . 7  |-  dom inl  =  _V
4744, 46syl6eleqr 2188 . . . . . 6  |-  ( ph  ->  A  e.  dom inl )
4847, 15jctil 306 . . . . 5  |-  ( ph  ->  ( Fun inl  /\  A  e. 
dom inl ) )
49 funfvima 5565 . . . . 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 5405 . . . 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 1185 . . 3  |-  ( ph  ->  ( ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) `  (inl `  A ) )  =  ( ( F  o.  `'inl ) `  (inl `  A ) ) )
532, 52syl5eq 2139 . 2  |-  ( ph  ->  (case ( F ,  G ) `  (inl `  A ) )  =  ( ( F  o.  `'inl ) `  (inl `  A ) ) )
54 f1ofn 5289 . . . 4  |-  ( `'inl
: ( { (/) }  X.  _V ) -1-1-onto-> _V  ->  `'inl 
Fn  ( { (/) }  X.  _V ) )
558, 54ax-mp 7 . . 3  |-  `'inl  Fn  ( { (/) }  X.  _V )
56 f1of 5288 . . . . . 6  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  -> inl : _V --> ( {
(/) }  X.  _V )
)
576, 56ax-mp 7 . . . . 5  |- inl : _V --> ( { (/) }  X.  _V )
5857a1i 9 . . . 4  |-  ( ph  -> inl : _V --> ( {
(/) }  X.  _V )
)
5958, 44ffvelrnd 5474 . . 3  |-  ( ph  ->  (inl `  A )  e.  ( { (/) }  X.  _V ) )
60 fvco2 5408 . . 3  |-  ( ( `'inl  Fn  ( { (/) }  X.  _V )  /\  (inl `  A )  e.  ( { (/) }  X.  _V ) )  ->  (
( F  o.  `'inl ) `  (inl `  A
) )  =  ( F `  ( `'inl `  (inl `  A )
) ) )
6155, 59, 60sylancr 406 . 2  |-  ( ph  ->  ( ( F  o.  `'inl ) `  (inl `  A ) )  =  ( F `  ( `'inl `  (inl `  A
) ) ) )
62 f1ocnvfv1 5594 . . . 4  |-  ( (inl : _V -1-1-onto-> ( { (/) }  X.  _V )  /\  A  e. 
_V )  ->  ( `'inl `  (inl `  A
) )  =  A )
636, 44, 62sylancr 406 . . 3  |-  ( ph  ->  ( `'inl `  (inl `  A ) )  =  A )
6463fveq2d 5344 . 2  |-  ( ph  ->  ( F `  ( `'inl `  (inl `  A
) ) )  =  ( F `  A
) )
6553, 61, 643eqtrd 2131 1  |-  ( ph  ->  (case ( F ,  G ) `  (inl `  A ) )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1296    e. wcel 1445   _Vcvv 2633    u. cun 3011    i^i cin 3012   (/)c0 3302   {csn 3466   <.cop 3469    X. cxp 4465   `'ccnv 4466   dom cdm 4467   "cima 4470    o. ccom 4471   Rel wrel 4472   Fun wfun 5043    Fn wfn 5044   -->wf 5045   -1-1-onto->wf1o 5048   ` cfv 5049   1oc1o 6212  inlcinl 6817  inrcinr 6818  casecdjucase 6854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950  df-1o 6219  df-inl 6819  df-inr 6820  df-case 6855
This theorem is referenced by:  ctm  6871
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