Theorem caseinr 7422

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 7414	. . . 4 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	fveq1i 5691	. . 3 |- (case ( F , G ) ` (inr ` A ) ) = ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inr ` A ) )
3		caseinr.f	. . . . . 6 |- ( ph -> Fun F )
4		djulf1o 7388	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
5		f1ocnv 5647	. . . . . . . 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 5636	. . . . . . 7 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> Fun `'inl )
8	6, 7	ax-mp 5	. . . . . 6 |- Fun `'inl
9		funco 5412	. . . . . 6 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
10	3, 8, 9	sylancl 417	. . . . 5 |- ( ph -> Fun ( F o. `'inl ) )
11		dmco 5291	. . . . . . 7 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
12		imacnvcnv 5247	. . . . . . 7 |- ( `' `'inl " dom F ) = (inl " dom F )
13	11, 12	eqtri 2259	. . . . . 6 |- dom ( F o. `'inl ) = (inl " dom F )
14	13	a1i 9	. . . . 5 |- ( ph -> dom ( F o. `'inl ) = (inl " dom F ) )
15		df-fn 5375	. . . . 5 |- ( ( F o. `'inl ) Fn (inl " dom F ) <-> ( Fun ( F o. `'inl ) /\ dom ( F o. `'inl ) = (inl " dom F ) ) )
16	10, 14, 15	sylanbrc 421	. . . 4 |- ( ph -> ( F o. `'inl ) Fn (inl " dom F ) )
17		caseinr.g	. . . . . . 7 |- ( ph -> G Fn B )
18		fnfun 5473	. . . . . . 7 |- ( G Fn B -> Fun G )
19	17, 18	syl 14	. . . . . 6 |- ( ph -> Fun G )
20		djurf1o 7389	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
21		f1ocnv 5647	. . . . . . . 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 5636	. . . . . . 7 |- ( `'inr : ( { 1o } X. _V ) -1-1-onto-> _V -> Fun `'inr )
24	22, 23	ax-mp 5	. . . . . 6 |- Fun `'inr
25		funco 5412	. . . . . 6 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
26	19, 24, 25	sylancl 417	. . . . 5 |- ( ph -> Fun ( G o. `'inr ) )
27		dmco 5291	. . . . . 6 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
28		df-inr 7378	. . . . . . . . . . 11 |- inr = ( x e. _V |-> <. 1o , x >. )
29	28	funmpt2 5411	. . . . . . . . . 10 |- Fun inr
30		funrel 5389	. . . . . . . . . 10 |- ( Fun inr -> Rel inr )
31	29, 30	ax-mp 5	. . . . . . . . 9 |- Rel inr
32		dfrel2 5233	. . . . . . . . 9 |- ( Rel inr <-> `' `'inr = inr )
33	31, 32	mpbi 145	. . . . . . . 8 |- `' `'inr = inr
34	33	a1i 9	. . . . . . 7 |- ( ph -> `' `'inr = inr )
35		fndm 5475	. . . . . . . 8 |- ( G Fn B -> dom G = B )
36	17, 35	syl 14	. . . . . . 7 |- ( ph -> dom G = B )
37	34, 36	imaeq12d 5122	. . . . . 6 |- ( ph -> ( `' `'inr " dom G ) = (inr " B ) )
38	27, 37	eqtrid 2283	. . . . 5 |- ( ph -> dom ( G o. `'inr ) = (inr " B ) )
39		df-fn 5375	. . . . 5 |- ( ( G o. `'inr ) Fn (inr " B ) <-> ( Fun ( G o. `'inr ) /\ dom ( G o. `'inr ) = (inr " B ) ) )
40	26, 38, 39	sylanbrc 421	. . . 4 |- ( ph -> ( G o. `'inr ) Fn (inr " B ) )
41		djuin 7394	. . . . 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 2835	. . . . . . 7 |- ( ph -> A e. _V )
45		f1odm 5638	. . . . . . . 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 2332	. . . . . 6 |- ( ph -> A e. dom inr )
48	47, 29	jctil 312	. . . . 5 |- ( ph -> ( Fun inr /\ A e. dom inr ) )
49		funfvima 5940	. . . . 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 5764	. . . 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 1282	. . 3 |- ( ph -> ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inr ` A ) ) = ( ( G o. `'inr ) ` (inr ` A ) ) )
53	2, 52	eqtrid 2283	. 2 |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( ( G o. `'inr ) ` (inr ` A ) ) )
54		f1ofn 5635	. . . 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 5634	. . . . . 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 5835	. . 3 |- ( ph -> (inr ` A ) e. ( { 1o } X. _V ) )
60		fvco2 5768	. . 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 418	. 2 |- ( ph -> ( ( G o. `'inr ) ` (inr ` A ) ) = ( G ` ( `'inr ` (inr ` A ) ) ) )
62		f1ocnvfv1 5973	. . . 4 |- ( (inr : _V -1-1-onto-> ( { 1o } X. _V ) /\ A e. _V ) -> ( `'inr ` (inr ` A ) ) = A )
63	20, 44, 62	sylancr 418	. . 3 |- ( ph -> ( `'inr ` (inr ` A ) ) = A )
64	63	fveq2d 5694	. 2 |- ( ph -> ( G ` ( `'inr ` (inr ` A ) ) ) = ( G ` A ) )
65	53, 61, 64	3eqtrd 2275	1 |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( G ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3705   <.cop 3708   X. cxp 4767   `'ccnv 4768   dom cdm 4769   "cima 4772   o. ccom 4773   Rel wrel 4774   Fun wfun 5366   Fn wfn 5367   -->wf 5368   -1-1-onto->wf1o 5371   ` cfv 5372   1oc1o 6670  inlcinl 7375  inrcinr 7376  casecdjucase 7413

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-inl 7377  df-inr 7378  df-case 7414

This theorem is referenced by:  omp1eomlem  7424