Theorem caseinl 6862

Description: Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)

Hypotheses

Ref	Expression
caseinl.f	|- ( ph -> F Fn B )
caseinl.g	|- ( ph -> Fun G )
caseinl.a	|- ( ph -> A e. B )

Assertion

Ref	Expression
caseinl	|- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( F ` A ) )

Proof of Theorem caseinl

Step	Hyp	Ref	Expression
1		df-case 6855	. . . 4 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	fveq1i 5341	. . 3 |- (case ( F , G ) ` (inl ` A ) ) = ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inl ` A ) )
3		caseinl.f	. . . . . . 7 |- ( ph -> F Fn B )
4		fnfun 5145	. . . . . . 7 |- ( F Fn B -> Fun F )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> Fun F )
6		djulf1o 6830	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
7		f1ocnv 5301	. . . . . . . 8 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> `'inl : ( { (/) } X. _V ) -1-1-onto-> _V )
8	6, 7	ax-mp 7	. . . . . . 7 |- `'inl : ( { (/) } X. _V ) -1-1-onto-> _V
9		f1ofun 5290	. . . . . . 7 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> Fun `'inl )
10	8, 9	ax-mp 7	. . . . . 6 |- Fun `'inl
11		funco 5088	. . . . . 6 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
12	5, 10, 11	sylancl 405	. . . . 5 |- ( ph -> Fun ( F o. `'inl ) )
13		dmco 4973	. . . . . 6 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
14		df-inl 6819	. . . . . . . . . . 11 |- inl = ( x e. _V |-> <. (/) , x >. )
15	14	funmpt2 5087	. . . . . . . . . 10 |- Fun inl
16		funrel 5066	. . . . . . . . . 10 |- ( Fun inl -> Rel inl )
17	15, 16	ax-mp 7	. . . . . . . . 9 |- Rel inl
18		dfrel2 4915	. . . . . . . . 9 |- ( Rel inl <-> `' `'inl = inl )
19	17, 18	mpbi 144	. . . . . . . 8 |- `' `'inl = inl
20	19	a1i 9	. . . . . . 7 |- ( ph -> `' `'inl = inl )
21		fndm 5147	. . . . . . . 8 |- ( F Fn B -> dom F = B )
22	3, 21	syl 14	. . . . . . 7 |- ( ph -> dom F = B )
23	20, 22	imaeq12d 4808	. . . . . 6 |- ( ph -> ( `' `'inl " dom F ) = (inl " B ) )
24	13, 23	syl5eq 2139	. . . . 5 |- ( ph -> dom ( F o. `'inl ) = (inl " B ) )
25		df-fn 5052	. . . . 5 |- ( ( F o. `'inl ) Fn (inl " B ) <-> ( Fun ( F o. `'inl ) /\ dom ( F o. `'inl ) = (inl " B ) ) )
26	12, 24, 25	sylanbrc 409	. . . 4 |- ( ph -> ( F o. `'inl ) Fn (inl " B ) )
27		caseinl.g	. . . . . 6 |- ( ph -> Fun G )
28		djurf1o 6831	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
29		f1ocnv 5301	. . . . . . . 8 |- (inr : _V -1-1-onto-> ( { 1o } X. _V ) -> `'inr : ( { 1o } X. _V ) -1-1-onto-> _V )
30	28, 29	ax-mp 7	. . . . . . 7 |- `'inr : ( { 1o } X. _V ) -1-1-onto-> _V
31		f1ofun 5290	. . . . . . 7 |- ( `'inr : ( { 1o } X. _V ) -1-1-onto-> _V -> Fun `'inr )
32	30, 31	ax-mp 7	. . . . . 6 |- Fun `'inr
33		funco 5088	. . . . . 6 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
34	27, 32, 33	sylancl 405	. . . . 5 |- ( ph -> Fun ( G o. `'inr ) )
35		dmco 4973	. . . . . . 7 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
36		imacnvcnv 4929	. . . . . . 7 |- ( `' `'inr " dom G ) = (inr " dom G )
37	35, 36	eqtri 2115	. . . . . 6 |- dom ( G o. `'inr ) = (inr " dom G )
38	37	a1i 9	. . . . 5 |- ( ph -> dom ( G o. `'inr ) = (inr " dom G ) )
39		df-fn 5052	. . . . 5 |- ( ( G o. `'inr ) Fn (inr " dom G ) <-> ( Fun ( G o. `'inr ) /\ dom ( G o. `'inr ) = (inr " dom G ) ) )
40	34, 38, 39	sylanbrc 409	. . . 4 |- ( ph -> ( G o. `'inr ) Fn (inr " dom G ) )
41		djuin 6836	. . . . 5 |- ( (inl " B ) i^i (inr " dom G ) ) = (/)
42	41	a1i 9	. . . 4 |- ( ph -> ( (inl " B ) i^i (inr " dom G ) ) = (/) )
43		caseinl.a	. . . . . . . 8 |- ( ph -> A e. B )
44	43	elexd 2646	. . . . . . 7 |- ( ph -> A e. _V )
45		f1odm 5292	. . . . . . . 8 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> dom inl = _V )
46	6, 45	ax-mp 7	. . . . . . 7 |- dom inl = _V
47	44, 46	syl6eleqr 2188	. . . . . 6 |- ( ph -> A e. dom inl )
48	47, 15	jctil 306	. . . . 5 |- ( ph -> ( Fun inl /\ A e. dom inl ) )
49		funfvima 5565	. . . . 5 |- ( ( Fun inl /\ A e. dom inl ) -> ( A e. B -> (inl ` A ) e. (inl " B ) ) )
50	48, 43, 49	sylc 62	. . . 4 |- ( ph -> (inl ` A ) e. (inl " B ) )
51		fvun1 5405	. . . 4 |- ( ( ( F o. `'inl ) Fn (inl " B ) /\ ( G o. `'inr ) Fn (inr " dom G ) /\ ( ( (inl " B ) i^i (inr " dom G ) ) = (/) /\ (inl ` A ) e. (inl " B ) ) ) -> ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inl ` A ) ) = ( ( F o. `'inl ) ` (inl ` A ) ) )
52	26, 40, 42, 50, 51	syl112anc 1185	. . 3 |- ( ph -> ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inl ` A ) ) = ( ( F o. `'inl ) ` (inl ` A ) ) )
53	2, 52	syl5eq 2139	. 2 |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( ( F o. `'inl ) ` (inl ` A ) ) )
54		f1ofn 5289	. . . 4 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> `'inl Fn ( { (/) } X. _V ) )
55	8, 54	ax-mp 7	. . 3 |- `'inl Fn ( { (/) } X. _V )
56		f1of 5288	. . . . . 6 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> inl : _V --> ( { (/) } X. _V ) )
57	6, 56	ax-mp 7	. . . . 5 |- inl : _V --> ( { (/) } X. _V )
58	57	a1i 9	. . . 4 |- ( ph -> inl : _V --> ( { (/) } X. _V ) )
59	58, 44	ffvelrnd 5474	. . 3 |- ( ph -> (inl ` A ) e. ( { (/) } X. _V ) )
60		fvco2 5408	. . 3 |- ( ( `'inl Fn ( { (/) } X. _V ) /\ (inl ` A ) e. ( { (/) } X. _V ) ) -> ( ( F o. `'inl ) ` (inl ` A ) ) = ( F ` ( `'inl ` (inl ` A ) ) ) )
61	55, 59, 60	sylancr 406	. 2 |- ( ph -> ( ( F o. `'inl ) ` (inl ` A ) ) = ( F ` ( `'inl ` (inl ` A ) ) ) )
62		f1ocnvfv1 5594	. . . 4 |- ( (inl : _V -1-1-onto-> ( { (/) } X. _V ) /\ A e. _V ) -> ( `'inl ` (inl ` A ) ) = A )
63	6, 44, 62	sylancr 406	. . 3 |- ( ph -> ( `'inl ` (inl ` A ) ) = A )
64	63	fveq2d 5344	. 2 |- ( ph -> ( F ` ( `'inl ` (inl ` A ) ) ) = ( F ` A ) )
65	53, 61, 64	3eqtrd 2131	1 |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1296   e. wcel 1445   _Vcvv 2633   u. cun 3011   i^i cin 3012   (/)c0 3302   {csn 3466   <.cop 3469   X. cxp 4465   `'ccnv 4466   dom cdm 4467   "cima 4470   o. ccom 4471   Rel wrel 4472   Fun wfun 5043   Fn wfn 5044   -->wf 5045   -1-1-onto->wf1o 5048   ` cfv 5049   1oc1o 6212  inlcinl 6817  inrcinr 6818  casecdjucase 6854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950  df-1o 6219  df-inl 6819  df-inr 6820  df-case 6855

This theorem is referenced by:  ctm  6871