Theorem caseinr 7293

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 7285	. . . 4 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	fveq1i 5640	. . 3 |- (case ( F , G ) ` (inr ` A ) ) = ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inr ` A ) )
3		caseinr.f	. . . . . 6 |- ( ph -> Fun F )
4		djulf1o 7259	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
5		f1ocnv 5596	. . . . . . . 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 5585	. . . . . . 7 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> Fun `'inl )
8	6, 7	ax-mp 5	. . . . . 6 |- Fun `'inl
9		funco 5365	. . . . . 6 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
10	3, 8, 9	sylancl 413	. . . . 5 |- ( ph -> Fun ( F o. `'inl ) )
11		dmco 5244	. . . . . . 7 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
12		imacnvcnv 5200	. . . . . . 7 |- ( `' `'inl " dom F ) = (inl " dom F )
13	11, 12	eqtri 2251	. . . . . 6 |- dom ( F o. `'inl ) = (inl " dom F )
14	13	a1i 9	. . . . 5 |- ( ph -> dom ( F o. `'inl ) = (inl " dom F ) )
15		df-fn 5328	. . . . 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 5426	. . . . . . 7 |- ( G Fn B -> Fun G )
19	17, 18	syl 14	. . . . . 6 |- ( ph -> Fun G )
20		djurf1o 7260	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
21		f1ocnv 5596	. . . . . . . 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 5585	. . . . . . 7 |- ( `'inr : ( { 1o } X. _V ) -1-1-onto-> _V -> Fun `'inr )
24	22, 23	ax-mp 5	. . . . . 6 |- Fun `'inr
25		funco 5365	. . . . . 6 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
26	19, 24, 25	sylancl 413	. . . . 5 |- ( ph -> Fun ( G o. `'inr ) )
27		dmco 5244	. . . . . 6 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
28		df-inr 7249	. . . . . . . . . . 11 |- inr = ( x e. _V |-> <. 1o , x >. )
29	28	funmpt2 5364	. . . . . . . . . 10 |- Fun inr
30		funrel 5342	. . . . . . . . . 10 |- ( Fun inr -> Rel inr )
31	29, 30	ax-mp 5	. . . . . . . . 9 |- Rel inr
32		dfrel2 5186	. . . . . . . . 9 |- ( Rel inr <-> `' `'inr = inr )
33	31, 32	mpbi 145	. . . . . . . 8 |- `' `'inr = inr
34	33	a1i 9	. . . . . . 7 |- ( ph -> `' `'inr = inr )
35		fndm 5428	. . . . . . . 8 |- ( G Fn B -> dom G = B )
36	17, 35	syl 14	. . . . . . 7 |- ( ph -> dom G = B )
37	34, 36	imaeq12d 5076	. . . . . 6 |- ( ph -> ( `' `'inr " dom G ) = (inr " B ) )
38	27, 37	eqtrid 2275	. . . . 5 |- ( ph -> dom ( G o. `'inr ) = (inr " B ) )
39		df-fn 5328	. . . . 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 7265	. . . . 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 2815	. . . . . . 7 |- ( ph -> A e. _V )
45		f1odm 5587	. . . . . . . 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 2324	. . . . . 6 |- ( ph -> A e. dom inr )
48	47, 29	jctil 312	. . . . 5 |- ( ph -> ( Fun inr /\ A e. dom inr ) )
49		funfvima 5888	. . . . 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 5713	. . . 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 1277	. . 3 |- ( ph -> ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inr ` A ) ) = ( ( G o. `'inr ) ` (inr ` A ) ) )
53	2, 52	eqtrid 2275	. 2 |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( ( G o. `'inr ) ` (inr ` A ) ) )
54		f1ofn 5584	. . . 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 5583	. . . . . 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 5783	. . 3 |- ( ph -> (inr ` A ) e. ( { 1o } X. _V ) )
60		fvco2 5715	. . 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 5920	. . . 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 5643	. 2 |- ( ph -> ( G ` ( `'inr ` (inr ` A ) ) ) = ( G ` A ) )
65	53, 61, 64	3eqtrd 2267	1 |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( G ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2201   _Vcvv 2801   u. cun 3197   i^i cin 3198   (/)c0 3493   {csn 3668   <.cop 3671   X. cxp 4722   `'ccnv 4723   dom cdm 4724   "cima 4727   o. ccom 4728   Rel wrel 4729   Fun wfun 5319   Fn wfn 5320   -->wf 5321   -1-1-onto->wf1o 5324   ` cfv 5325   1oc1o 6577  inlcinl 7246  inrcinr 7247  casecdjucase 7284

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-id 4389  df-iord 4462  df-on 4464  df-suc 4467  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-1st 6305  df-2nd 6306  df-1o 6584  df-inl 7248  df-inr 7249  df-case 7285

This theorem is referenced by:  omp1eomlem  7295