Theorem caseinr 6977

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

Step	Hyp	Ref	Expression
1		df-case 6969	. . . 4 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	fveq1i 5422	. . 3 |- (case ( F , G ) ` (inr ` A ) ) = ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inr ` A ) )
3		caseinr.f	. . . . . 6 |- ( ph -> Fun F )
4		djulf1o 6943	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
5		f1ocnv 5380	. . . . . . . 8 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> `'inl : ( { (/) } X. _V ) -1-1-onto-> _V )
6	4, 5	ax-mp 5	. . . . . . 7 |- `'inl : ( { (/) } X. _V ) -1-1-onto-> _V
7		f1ofun 5369	. . . . . . 7 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> Fun `'inl )
8	6, 7	ax-mp 5	. . . . . 6 |- Fun `'inl
9		funco 5163	. . . . . 6 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
10	3, 8, 9	sylancl 409	. . . . 5 |- ( ph -> Fun ( F o. `'inl ) )
11		dmco 5047	. . . . . . 7 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
12		imacnvcnv 5003	. . . . . . 7 |- ( `' `'inl " dom F ) = (inl " dom F )
13	11, 12	eqtri 2160	. . . . . 6 |- dom ( F o. `'inl ) = (inl " dom F )
14	13	a1i 9	. . . . 5 |- ( ph -> dom ( F o. `'inl ) = (inl " dom F ) )
15		df-fn 5126	. . . . 5 |- ( ( F o. `'inl ) Fn (inl " dom F ) <-> ( Fun ( F o. `'inl ) /\ dom ( F o. `'inl ) = (inl " dom F ) ) )
16	10, 14, 15	sylanbrc 413	. . . 4 |- ( ph -> ( F o. `'inl ) Fn (inl " dom F ) )
17		caseinr.g	. . . . . . 7 |- ( ph -> G Fn B )
18		fnfun 5220	. . . . . . 7 |- ( G Fn B -> Fun G )
19	17, 18	syl 14	. . . . . 6 |- ( ph -> Fun G )
20		djurf1o 6944	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
21		f1ocnv 5380	. . . . . . . 8 |- (inr : _V -1-1-onto-> ( { 1o } X. _V ) -> `'inr : ( { 1o } X. _V ) -1-1-onto-> _V )
22	20, 21	ax-mp 5	. . . . . . 7 |- `'inr : ( { 1o } X. _V ) -1-1-onto-> _V
23		f1ofun 5369	. . . . . . 7 |- ( `'inr : ( { 1o } X. _V ) -1-1-onto-> _V -> Fun `'inr )
24	22, 23	ax-mp 5	. . . . . 6 |- Fun `'inr
25		funco 5163	. . . . . 6 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
26	19, 24, 25	sylancl 409	. . . . 5 |- ( ph -> Fun ( G o. `'inr ) )
27		dmco 5047	. . . . . 6 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
28		df-inr 6933	. . . . . . . . . . 11 |- inr = ( x e. _V |-> <. 1o , x >. )
29	28	funmpt2 5162	. . . . . . . . . 10 |- Fun inr
30		funrel 5140	. . . . . . . . . 10 |- ( Fun inr -> Rel inr )
31	29, 30	ax-mp 5	. . . . . . . . 9 |- Rel inr
32		dfrel2 4989	. . . . . . . . 9 |- ( Rel inr <-> `' `'inr = inr )
33	31, 32	mpbi 144	. . . . . . . 8 |- `' `'inr = inr
34	33	a1i 9	. . . . . . 7 |- ( ph -> `' `'inr = inr )
35		fndm 5222	. . . . . . . 8 |- ( G Fn B -> dom G = B )
36	17, 35	syl 14	. . . . . . 7 |- ( ph -> dom G = B )
37	34, 36	imaeq12d 4882	. . . . . 6 |- ( ph -> ( `' `'inr " dom G ) = (inr " B ) )
38	27, 37	syl5eq 2184	. . . . 5 |- ( ph -> dom ( G o. `'inr ) = (inr " B ) )
39		df-fn 5126	. . . . 5 |- ( ( G o. `'inr ) Fn (inr " B ) <-> ( Fun ( G o. `'inr ) /\ dom ( G o. `'inr ) = (inr " B ) ) )
40	26, 38, 39	sylanbrc 413	. . . 4 |- ( ph -> ( G o. `'inr ) Fn (inr " B ) )
41		djuin 6949	. . . . 5 |- ( (inl " dom F ) i^i (inr " B ) ) = (/)
42	41	a1i 9	. . . 4 |- ( ph -> ( (inl " dom F ) i^i (inr " B ) ) = (/) )
43		caseinr.a	. . . . . . . 8 |- ( ph -> A e. B )
44	43	elexd 2699	. . . . . . 7 |- ( ph -> A e. _V )
45		f1odm 5371	. . . . . . . 8 |- (inr : _V -1-1-onto-> ( { 1o } X. _V ) -> dom inr = _V )
46	20, 45	ax-mp 5	. . . . . . 7 |- dom inr = _V
47	44, 46	eleqtrrdi 2233	. . . . . 6 |- ( ph -> A e. dom inr )
48	47, 29	jctil 310	. . . . 5 |- ( ph -> ( Fun inr /\ A e. dom inr ) )
49		funfvima 5649	. . . . 5 |- ( ( Fun inr /\ A e. dom inr ) -> ( A e. B -> (inr ` A ) e. (inr " B ) ) )
50	48, 43, 49	sylc 62	. . . 4 |- ( ph -> (inr ` A ) e. (inr " B ) )
51		fvun2 5488	. . . 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 ) ) )
52	16, 40, 42, 50, 51	syl112anc 1220	. . 3 |- ( ph -> ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inr ` A ) ) = ( ( G o. `'inr ) ` (inr ` A ) ) )
53	2, 52	syl5eq 2184	. 2 |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( ( G o. `'inr ) ` (inr ` A ) ) )
54		f1ofn 5368	. . . 4 |- ( `'inr : ( { 1o } X. _V ) -1-1-onto-> _V -> `'inr Fn ( { 1o } X. _V ) )
55	22, 54	ax-mp 5	. . 3 |- `'inr Fn ( { 1o } X. _V )
56		f1of 5367	. . . . . 6 |- (inr : _V -1-1-onto-> ( { 1o } X. _V ) -> inr : _V --> ( { 1o } X. _V ) )
57	20, 56	ax-mp 5	. . . . 5 |- inr : _V --> ( { 1o } X. _V )
58	57	a1i 9	. . . 4 |- ( ph -> inr : _V --> ( { 1o } X. _V ) )
59	58, 44	ffvelrnd 5556	. . 3 |- ( ph -> (inr ` A ) e. ( { 1o } X. _V ) )
60		fvco2 5490	. . 3 |- ( ( `'inr Fn ( { 1o } X. _V ) /\ (inr ` A ) e. ( { 1o } X. _V ) ) -> ( ( G o. `'inr ) ` (inr ` A ) ) = ( G ` ( `'inr ` (inr ` A ) ) ) )
61	55, 59, 60	sylancr 410	. 2 |- ( ph -> ( ( G o. `'inr ) ` (inr ` A ) ) = ( G ` ( `'inr ` (inr ` A ) ) ) )
62		f1ocnvfv1 5678	. . . 4 |- ( (inr : _V -1-1-onto-> ( { 1o } X. _V ) /\ A e. _V ) -> ( `'inr ` (inr ` A ) ) = A )
63	20, 44, 62	sylancr 410	. . 3 |- ( ph -> ( `'inr ` (inr ` A ) ) = A )
64	63	fveq2d 5425	. 2 |- ( ph -> ( G ` ( `'inr ` (inr ` A ) ) ) = ( G ` A ) )
65	53, 61, 64	3eqtrd 2176	1 |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( G ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   u. cun 3069   i^i cin 3070   (/)c0 3363   {csn 3527   <.cop 3530   X. cxp 4537   `'ccnv 4538   dom cdm 4539   "cima 4542   o. ccom 4543   Rel wrel 4544   Fun wfun 5117   Fn wfn 5118   -->wf 5119   -1-1-onto->wf1o 5122   ` cfv 5123   1oc1o 6306  inlcinl 6930  inrcinr 6931  casecdjucase 6968

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 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-inl 6932  df-inr 6933  df-case 6969

This theorem is referenced by:  omp1eomlem  6979