Theorem caseinl 7150

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 7143	. . . 4 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	fveq1i 5555	. . 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 5351	. . . . . . 7 |- ( F Fn B -> Fun F )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> Fun F )
6		djulf1o 7117	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
7		f1ocnv 5513	. . . . . . . 8 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> `'inl : ( { (/) } X. _V ) -1-1-onto-> _V )
8	6, 7	ax-mp 5	. . . . . . 7 |- `'inl : ( { (/) } X. _V ) -1-1-onto-> _V
9		f1ofun 5502	. . . . . . 7 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> Fun `'inl )
10	8, 9	ax-mp 5	. . . . . 6 |- Fun `'inl
11		funco 5294	. . . . . 6 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
12	5, 10, 11	sylancl 413	. . . . 5 |- ( ph -> Fun ( F o. `'inl ) )
13		dmco 5174	. . . . . 6 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
14		df-inl 7106	. . . . . . . . . . 11 |- inl = ( x e. _V |-> <. (/) , x >. )
15	14	funmpt2 5293	. . . . . . . . . 10 |- Fun inl
16		funrel 5271	. . . . . . . . . 10 |- ( Fun inl -> Rel inl )
17	15, 16	ax-mp 5	. . . . . . . . 9 |- Rel inl
18		dfrel2 5116	. . . . . . . . 9 |- ( Rel inl <-> `' `'inl = inl )
19	17, 18	mpbi 145	. . . . . . . 8 |- `' `'inl = inl
20	19	a1i 9	. . . . . . 7 |- ( ph -> `' `'inl = inl )
21		fndm 5353	. . . . . . . 8 |- ( F Fn B -> dom F = B )
22	3, 21	syl 14	. . . . . . 7 |- ( ph -> dom F = B )
23	20, 22	imaeq12d 5006	. . . . . 6 |- ( ph -> ( `' `'inl " dom F ) = (inl " B ) )
24	13, 23	eqtrid 2238	. . . . 5 |- ( ph -> dom ( F o. `'inl ) = (inl " B ) )
25		df-fn 5257	. . . . 5 |- ( ( F o. `'inl ) Fn (inl " B ) <-> ( Fun ( F o. `'inl ) /\ dom ( F o. `'inl ) = (inl " B ) ) )
26	12, 24, 25	sylanbrc 417	. . . 4 |- ( ph -> ( F o. `'inl ) Fn (inl " B ) )
27		caseinl.g	. . . . . 6 |- ( ph -> Fun G )
28		djurf1o 7118	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
29		f1ocnv 5513	. . . . . . . 8 |- (inr : _V -1-1-onto-> ( { 1o } X. _V ) -> `'inr : ( { 1o } X. _V ) -1-1-onto-> _V )
30	28, 29	ax-mp 5	. . . . . . 7 |- `'inr : ( { 1o } X. _V ) -1-1-onto-> _V
31		f1ofun 5502	. . . . . . 7 |- ( `'inr : ( { 1o } X. _V ) -1-1-onto-> _V -> Fun `'inr )
32	30, 31	ax-mp 5	. . . . . 6 |- Fun `'inr
33		funco 5294	. . . . . 6 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
34	27, 32, 33	sylancl 413	. . . . 5 |- ( ph -> Fun ( G o. `'inr ) )
35		dmco 5174	. . . . . . 7 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
36		imacnvcnv 5130	. . . . . . 7 |- ( `' `'inr " dom G ) = (inr " dom G )
37	35, 36	eqtri 2214	. . . . . 6 |- dom ( G o. `'inr ) = (inr " dom G )
38	37	a1i 9	. . . . 5 |- ( ph -> dom ( G o. `'inr ) = (inr " dom G ) )
39		df-fn 5257	. . . . 5 |- ( ( G o. `'inr ) Fn (inr " dom G ) <-> ( Fun ( G o. `'inr ) /\ dom ( G o. `'inr ) = (inr " dom G ) ) )
40	34, 38, 39	sylanbrc 417	. . . 4 |- ( ph -> ( G o. `'inr ) Fn (inr " dom G ) )
41		djuin 7123	. . . . 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 2773	. . . . . . 7 |- ( ph -> A e. _V )
45		f1odm 5504	. . . . . . . 8 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> dom inl = _V )
46	6, 45	ax-mp 5	. . . . . . 7 |- dom inl = _V
47	44, 46	eleqtrrdi 2287	. . . . . 6 |- ( ph -> A e. dom inl )
48	47, 15	jctil 312	. . . . 5 |- ( ph -> ( Fun inl /\ A e. dom inl ) )
49		funfvima 5790	. . . . 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 5623	. . . 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 1253	. . 3 |- ( ph -> ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inl ` A ) ) = ( ( F o. `'inl ) ` (inl ` A ) ) )
53	2, 52	eqtrid 2238	. 2 |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( ( F o. `'inl ) ` (inl ` A ) ) )
54		f1ofn 5501	. . . 4 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> `'inl Fn ( { (/) } X. _V ) )
55	8, 54	ax-mp 5	. . 3 |- `'inl Fn ( { (/) } X. _V )
56		f1of 5500	. . . . . 6 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> inl : _V --> ( { (/) } X. _V ) )
57	6, 56	ax-mp 5	. . . . 5 |- inl : _V --> ( { (/) } X. _V )
58	57	a1i 9	. . . 4 |- ( ph -> inl : _V --> ( { (/) } X. _V ) )
59	58, 44	ffvelcdmd 5694	. . 3 |- ( ph -> (inl ` A ) e. ( { (/) } X. _V ) )
60		fvco2 5626	. . 3 |- ( ( `'inl Fn ( { (/) } X. _V ) /\ (inl ` A ) e. ( { (/) } X. _V ) ) -> ( ( F o. `'inl ) ` (inl ` A ) ) = ( F ` ( `'inl ` (inl ` A ) ) ) )
61	55, 59, 60	sylancr 414	. 2 |- ( ph -> ( ( F o. `'inl ) ` (inl ` A ) ) = ( F ` ( `'inl ` (inl ` A ) ) ) )
62		f1ocnvfv1 5820	. . . 4 |- ( (inl : _V -1-1-onto-> ( { (/) } X. _V ) /\ A e. _V ) -> ( `'inl ` (inl ` A ) ) = A )
63	6, 44, 62	sylancr 414	. . 3 |- ( ph -> ( `'inl ` (inl ` A ) ) = A )
64	63	fveq2d 5558	. 2 |- ( ph -> ( F ` ( `'inl ` (inl ` A ) ) ) = ( F ` A ) )
65	53, 61, 64	3eqtrd 2230	1 |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   u. cun 3151   i^i cin 3152   (/)c0 3446   {csn 3618   <.cop 3621   X. cxp 4657   `'ccnv 4658   dom cdm 4659   "cima 4662   o. ccom 4663   Rel wrel 4664   Fun wfun 5248   Fn wfn 5249   -->wf 5250   -1-1-onto->wf1o 5253   ` cfv 5254   1oc1o 6462  inlcinl 7104  inrcinr 7105  casecdjucase 7142

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-inl 7106  df-inr 7107  df-case 7143

This theorem is referenced by:  omp1eomlem  7153  ctm  7168