Theorem caseinr 7167

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 7159	. . . 4 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	fveq1i 5562	. . 3 |- (case ( F , G ) ` (inr ` A ) ) = ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inr ` A ) )
3		caseinr.f	. . . . . 6 |- ( ph -> Fun F )
4		djulf1o 7133	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
5		f1ocnv 5520	. . . . . . . 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 5509	. . . . . . 7 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> Fun `'inl )
8	6, 7	ax-mp 5	. . . . . 6 |- Fun `'inl
9		funco 5299	. . . . . 6 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
10	3, 8, 9	sylancl 413	. . . . 5 |- ( ph -> Fun ( F o. `'inl ) )
11		dmco 5179	. . . . . . 7 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
12		imacnvcnv 5135	. . . . . . 7 |- ( `' `'inl " dom F ) = (inl " dom F )
13	11, 12	eqtri 2217	. . . . . 6 |- dom ( F o. `'inl ) = (inl " dom F )
14	13	a1i 9	. . . . 5 |- ( ph -> dom ( F o. `'inl ) = (inl " dom F ) )
15		df-fn 5262	. . . . 5 |- ( ( F o. `'inl ) Fn (inl " dom F ) <-> ( Fun ( F o. `'inl ) /\ dom ( F o. `'inl ) = (inl " dom F ) ) )
16	10, 14, 15	sylanbrc 417	. . . 4 |- ( ph -> ( F o. `'inl ) Fn (inl " dom F ) )
17		caseinr.g	. . . . . . 7 |- ( ph -> G Fn B )
18		fnfun 5356	. . . . . . 7 |- ( G Fn B -> Fun G )
19	17, 18	syl 14	. . . . . 6 |- ( ph -> Fun G )
20		djurf1o 7134	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
21		f1ocnv 5520	. . . . . . . 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 5509	. . . . . . 7 |- ( `'inr : ( { 1o } X. _V ) -1-1-onto-> _V -> Fun `'inr )
24	22, 23	ax-mp 5	. . . . . 6 |- Fun `'inr
25		funco 5299	. . . . . 6 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
26	19, 24, 25	sylancl 413	. . . . 5 |- ( ph -> Fun ( G o. `'inr ) )
27		dmco 5179	. . . . . 6 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
28		df-inr 7123	. . . . . . . . . . 11 |- inr = ( x e. _V |-> <. 1o , x >. )
29	28	funmpt2 5298	. . . . . . . . . 10 |- Fun inr
30		funrel 5276	. . . . . . . . . 10 |- ( Fun inr -> Rel inr )
31	29, 30	ax-mp 5	. . . . . . . . 9 |- Rel inr
32		dfrel2 5121	. . . . . . . . 9 |- ( Rel inr <-> `' `'inr = inr )
33	31, 32	mpbi 145	. . . . . . . 8 |- `' `'inr = inr
34	33	a1i 9	. . . . . . 7 |- ( ph -> `' `'inr = inr )
35		fndm 5358	. . . . . . . 8 |- ( G Fn B -> dom G = B )
36	17, 35	syl 14	. . . . . . 7 |- ( ph -> dom G = B )
37	34, 36	imaeq12d 5011	. . . . . 6 |- ( ph -> ( `' `'inr " dom G ) = (inr " B ) )
38	27, 37	eqtrid 2241	. . . . 5 |- ( ph -> dom ( G o. `'inr ) = (inr " B ) )
39		df-fn 5262	. . . . 5 |- ( ( G o. `'inr ) Fn (inr " B ) <-> ( Fun ( G o. `'inr ) /\ dom ( G o. `'inr ) = (inr " B ) ) )
40	26, 38, 39	sylanbrc 417	. . . 4 |- ( ph -> ( G o. `'inr ) Fn (inr " B ) )
41		djuin 7139	. . . . 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 2776	. . . . . . 7 |- ( ph -> A e. _V )
45		f1odm 5511	. . . . . . . 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 2290	. . . . . 6 |- ( ph -> A e. dom inr )
48	47, 29	jctil 312	. . . . 5 |- ( ph -> ( Fun inr /\ A e. dom inr ) )
49		funfvima 5797	. . . . 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 5631	. . . 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 1253	. . 3 |- ( ph -> ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inr ` A ) ) = ( ( G o. `'inr ) ` (inr ` A ) ) )
53	2, 52	eqtrid 2241	. 2 |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( ( G o. `'inr ) ` (inr ` A ) ) )
54		f1ofn 5508	. . . 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 5507	. . . . . 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	ffvelcdmd 5701	. . 3 |- ( ph -> (inr ` A ) e. ( { 1o } X. _V ) )
60		fvco2 5633	. . 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 414	. 2 |- ( ph -> ( ( G o. `'inr ) ` (inr ` A ) ) = ( G ` ( `'inr ` (inr ` A ) ) ) )
62		f1ocnvfv1 5827	. . . 4 |- ( (inr : _V -1-1-onto-> ( { 1o } X. _V ) /\ A e. _V ) -> ( `'inr ` (inr ` A ) ) = A )
63	20, 44, 62	sylancr 414	. . 3 |- ( ph -> ( `'inr ` (inr ` A ) ) = A )
64	63	fveq2d 5565	. 2 |- ( ph -> ( G ` ( `'inr ` (inr ` A ) ) ) = ( G ` A ) )
65	53, 61, 64	3eqtrd 2233	1 |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( G ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   u. cun 3155   i^i cin 3156   (/)c0 3451   {csn 3623   <.cop 3626   X. cxp 4662   `'ccnv 4663   dom cdm 4664   "cima 4667   o. ccom 4668   Rel wrel 4669   Fun wfun 5253   Fn wfn 5254   -->wf 5255   -1-1-onto->wf1o 5258   ` cfv 5259   1oc1o 6476  inlcinl 7120  inrcinr 7121  casecdjucase 7158

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1st 6207  df-2nd 6208  df-1o 6483  df-inl 7122  df-inr 7123  df-case 7159

This theorem is referenced by:  omp1eomlem  7169