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Theorem caseinr 6943
Description: Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
Hypotheses
Ref Expression
caseinr.f  |-  ( ph  ->  Fun  F )
caseinr.g  |-  ( ph  ->  G  Fn  B )
caseinr.a  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
caseinr  |-  ( ph  ->  (case ( F ,  G ) `  (inr `  A ) )  =  ( G `  A
) )

Proof of Theorem caseinr
StepHypRef Expression
1 df-case 6935 . . . 4  |- case ( F ,  G )  =  ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
)
21fveq1i 5388 . . 3  |-  (case ( F ,  G ) `
 (inr `  A
) )  =  ( ( ( F  o.  `'inl )  u.  ( G  o.  `'inr )
) `  (inr `  A
) )
3 caseinr.f . . . . . 6  |-  ( ph  ->  Fun  F )
4 djulf1o 6909 . . . . . . . 8  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
5 f1ocnv 5346 . . . . . . . 8  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  ->  `'inl : ( { (/) }  X.  _V ) -1-1-onto-> _V )
64, 5ax-mp 5 . . . . . . 7  |-  `'inl :
( { (/) }  X.  _V ) -1-1-onto-> _V
7 f1ofun 5335 . . . . . . 7  |-  ( `'inl
: ( { (/) }  X.  _V ) -1-1-onto-> _V  ->  Fun  `'inl )
86, 7ax-mp 5 . . . . . 6  |-  Fun  `'inl
9 funco 5131 . . . . . 6  |-  ( ( Fun  F  /\  Fun  `'inl )  ->  Fun  ( F  o.  `'inl ) )
103, 8, 9sylancl 407 . . . . 5  |-  ( ph  ->  Fun  ( F  o.  `'inl ) )
11 dmco 5015 . . . . . . 7  |-  dom  ( F  o.  `'inl )  =  ( `' `'inl " dom  F )
12 imacnvcnv 4971 . . . . . . 7  |-  ( `' `'inl " dom  F )  =  (inl " dom  F )
1311, 12eqtri 2136 . . . . . 6  |-  dom  ( F  o.  `'inl )  =  (inl " dom  F
)
1413a1i 9 . . . . 5  |-  ( ph  ->  dom  ( F  o.  `'inl )  =  (inl " dom  F ) )
15 df-fn 5094 . . . . 5  |-  ( ( F  o.  `'inl )  Fn  (inl " dom  F
)  <->  ( Fun  ( F  o.  `'inl )  /\  dom  ( F  o.  `'inl )  =  (inl " dom  F ) ) )
1610, 14, 15sylanbrc 411 . . . 4  |-  ( ph  ->  ( F  o.  `'inl )  Fn  (inl " dom  F ) )
17 caseinr.g . . . . . . 7  |-  ( ph  ->  G  Fn  B )
18 fnfun 5188 . . . . . . 7  |-  ( G  Fn  B  ->  Fun  G )
1917, 18syl 14 . . . . . 6  |-  ( ph  ->  Fun  G )
20 djurf1o 6910 . . . . . . . 8  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
21 f1ocnv 5346 . . . . . . . 8  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  ->  `'inr :
( { 1o }  X.  _V ) -1-1-onto-> _V )
2220, 21ax-mp 5 . . . . . . 7  |-  `'inr :
( { 1o }  X.  _V ) -1-1-onto-> _V
23 f1ofun 5335 . . . . . . 7  |-  ( `'inr
: ( { 1o }  X.  _V ) -1-1-onto-> _V  ->  Fun  `'inr )
2422, 23ax-mp 5 . . . . . 6  |-  Fun  `'inr
25 funco 5131 . . . . . 6  |-  ( ( Fun  G  /\  Fun  `'inr )  ->  Fun  ( G  o.  `'inr ) )
2619, 24, 25sylancl 407 . . . . 5  |-  ( ph  ->  Fun  ( G  o.  `'inr ) )
27 dmco 5015 . . . . . 6  |-  dom  ( G  o.  `'inr )  =  ( `' `'inr " dom  G )
28 df-inr 6899 . . . . . . . . . . 11  |- inr  =  ( x  e.  _V  |->  <. 1o ,  x >. )
2928funmpt2 5130 . . . . . . . . . 10  |-  Fun inr
30 funrel 5108 . . . . . . . . . 10  |-  ( Fun inr  ->  Rel inr )
3129, 30ax-mp 5 . . . . . . . . 9  |-  Rel inr
32 dfrel2 4957 . . . . . . . . 9  |-  ( Rel inr  <->  `' `'inr  = inr )
3331, 32mpbi 144 . . . . . . . 8  |-  `' `'inr  = inr
3433a1i 9 . . . . . . 7  |-  ( ph  ->  `' `'inr  = inr )
35 fndm 5190 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
3617, 35syl 14 . . . . . . 7  |-  ( ph  ->  dom  G  =  B )
3734, 36imaeq12d 4850 . . . . . 6  |-  ( ph  ->  ( `' `'inr " dom  G )  =  (inr " B ) )
3827, 37syl5eq 2160 . . . . 5  |-  ( ph  ->  dom  ( G  o.  `'inr )  =  (inr " B ) )
39 df-fn 5094 . . . . 5  |-  ( ( G  o.  `'inr )  Fn  (inr " B )  <-> 
( Fun  ( G  o.  `'inr )  /\  dom  ( G  o.  `'inr )  =  (inr " B ) ) )
4026, 38, 39sylanbrc 411 . . . 4  |-  ( ph  ->  ( G  o.  `'inr )  Fn  (inr " B
) )
41 djuin 6915 . . . . 5  |-  ( (inl " dom  F )  i^i  (inr " B ) )  =  (/)
4241a1i 9 . . . 4  |-  ( ph  ->  ( (inl " dom  F )  i^i  (inr " B ) )  =  (/) )
43 caseinr.a . . . . . . . 8  |-  ( ph  ->  A  e.  B )
4443elexd 2671 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
45 f1odm 5337 . . . . . . . 8  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  ->  dom inr  =  _V )
4620, 45ax-mp 5 . . . . . . 7  |-  dom inr  =  _V
4744, 46syl6eleqr 2209 . . . . . 6  |-  ( ph  ->  A  e.  dom inr )
4847, 29jctil 308 . . . . 5  |-  ( ph  ->  ( Fun inr  /\  A  e. 
dom inr ) )
49 funfvima 5615 . . . . 5  |-  ( ( Fun inr  /\  A  e.  dom inr )  ->  ( A  e.  B  ->  (inr `  A )  e.  (inr " B ) ) )
5048, 43, 49sylc 62 . . . 4  |-  ( ph  ->  (inr `  A )  e.  (inr " B ) )
51 fvun2 5454 . . . 4  |-  ( ( ( F  o.  `'inl )  Fn  (inl " dom  F )  /\  ( G  o.  `'inr )  Fn  (inr " B )  /\  ( ( (inl " dom  F )  i^i  (inr " B ) )  =  (/)  /\  (inr `  A )  e.  (inr " B ) ) )  ->  ( ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) `  (inr `  A ) )  =  ( ( G  o.  `'inr ) `  (inr `  A ) ) )
5216, 40, 42, 50, 51syl112anc 1203 . . 3  |-  ( ph  ->  ( ( ( F  o.  `'inl )  u.  ( G  o.  `'inr ) ) `  (inr `  A ) )  =  ( ( G  o.  `'inr ) `  (inr `  A ) ) )
532, 52syl5eq 2160 . 2  |-  ( ph  ->  (case ( F ,  G ) `  (inr `  A ) )  =  ( ( G  o.  `'inr ) `  (inr `  A ) ) )
54 f1ofn 5334 . . . 4  |-  ( `'inr
: ( { 1o }  X.  _V ) -1-1-onto-> _V  ->  `'inr 
Fn  ( { 1o }  X.  _V ) )
5522, 54ax-mp 5 . . 3  |-  `'inr  Fn  ( { 1o }  X.  _V )
56 f1of 5333 . . . . . 6  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  -> inr : _V --> ( { 1o }  X.  _V ) )
5720, 56ax-mp 5 . . . . 5  |- inr : _V --> ( { 1o }  X.  _V )
5857a1i 9 . . . 4  |-  ( ph  -> inr : _V --> ( { 1o }  X.  _V ) )
5958, 44ffvelrnd 5522 . . 3  |-  ( ph  ->  (inr `  A )  e.  ( { 1o }  X.  _V ) )
60 fvco2 5456 . . 3  |-  ( ( `'inr  Fn  ( { 1o }  X.  _V )  /\  (inr `  A )  e.  ( { 1o }  X.  _V ) )  -> 
( ( G  o.  `'inr ) `  (inr `  A ) )  =  ( G `  ( `'inr `  (inr `  A
) ) ) )
6155, 59, 60sylancr 408 . 2  |-  ( ph  ->  ( ( G  o.  `'inr ) `  (inr `  A ) )  =  ( G `  ( `'inr `  (inr `  A
) ) ) )
62 f1ocnvfv1 5644 . . . 4  |-  ( (inr : _V -1-1-onto-> ( { 1o }  X.  _V )  /\  A  e.  _V )  ->  ( `'inr `  (inr `  A
) )  =  A )
6320, 44, 62sylancr 408 . . 3  |-  ( ph  ->  ( `'inr `  (inr `  A ) )  =  A )
6463fveq2d 5391 . 2  |-  ( ph  ->  ( G `  ( `'inr `  (inr `  A
) ) )  =  ( G `  A
) )
6553, 61, 643eqtrd 2152 1  |-  ( ph  ->  (case ( F ,  G ) `  (inr `  A ) )  =  ( G `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   _Vcvv 2658    u. cun 3037    i^i cin 3038   (/)c0 3331   {csn 3495   <.cop 3498    X. cxp 4505   `'ccnv 4506   dom cdm 4507   "cima 4510    o. ccom 4511   Rel wrel 4512   Fun wfun 5085    Fn wfn 5086   -->wf 5087   -1-1-onto->wf1o 5090   ` cfv 5091   1oc1o 6272  inlcinl 6896  inrcinr 6897  casecdjucase 6934
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-inl 6898  df-inr 6899  df-case 6935
This theorem is referenced by:  omp1eomlem  6945
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