Theorem caseinl 7424

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 7417	. . . 4 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	fveq1i 5694	. . 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 5476	. . . . . . 7 |- ( F Fn B -> Fun F )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> Fun F )
6		djulf1o 7391	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
7		f1ocnv 5650	. . . . . . . 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 5639	. . . . . . 7 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> Fun `'inl )
10	8, 9	ax-mp 5	. . . . . 6 |- Fun `'inl
11		funco 5415	. . . . . 6 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
12	5, 10, 11	sylancl 417	. . . . 5 |- ( ph -> Fun ( F o. `'inl ) )
13		dmco 5294	. . . . . 6 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
14		df-inl 7380	. . . . . . . . . . 11 |- inl = ( x e. _V |-> <. (/) , x >. )
15	14	funmpt2 5414	. . . . . . . . . 10 |- Fun inl
16		funrel 5392	. . . . . . . . . 10 |- ( Fun inl -> Rel inl )
17	15, 16	ax-mp 5	. . . . . . . . 9 |- Rel inl
18		dfrel2 5236	. . . . . . . . 9 |- ( Rel inl <-> `' `'inl = inl )
19	17, 18	mpbi 145	. . . . . . . 8 |- `' `'inl = inl
20	19	a1i 9	. . . . . . 7 |- ( ph -> `' `'inl = inl )
21		fndm 5478	. . . . . . . 8 |- ( F Fn B -> dom F = B )
22	3, 21	syl 14	. . . . . . 7 |- ( ph -> dom F = B )
23	20, 22	imaeq12d 5125	. . . . . 6 |- ( ph -> ( `' `'inl " dom F ) = (inl " B ) )
24	13, 23	eqtrid 2283	. . . . 5 |- ( ph -> dom ( F o. `'inl ) = (inl " B ) )
25		df-fn 5378	. . . . 5 |- ( ( F o. `'inl ) Fn (inl " B ) <-> ( Fun ( F o. `'inl ) /\ dom ( F o. `'inl ) = (inl " B ) ) )
26	12, 24, 25	sylanbrc 421	. . . 4 |- ( ph -> ( F o. `'inl ) Fn (inl " B ) )
27		caseinl.g	. . . . . 6 |- ( ph -> Fun G )
28		djurf1o 7392	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
29		f1ocnv 5650	. . . . . . . 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 5639	. . . . . . 7 |- ( `'inr : ( { 1o } X. _V ) -1-1-onto-> _V -> Fun `'inr )
32	30, 31	ax-mp 5	. . . . . 6 |- Fun `'inr
33		funco 5415	. . . . . 6 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
34	27, 32, 33	sylancl 417	. . . . 5 |- ( ph -> Fun ( G o. `'inr ) )
35		dmco 5294	. . . . . . 7 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
36		imacnvcnv 5250	. . . . . . 7 |- ( `' `'inr " dom G ) = (inr " dom G )
37	35, 36	eqtri 2259	. . . . . 6 |- dom ( G o. `'inr ) = (inr " dom G )
38	37	a1i 9	. . . . 5 |- ( ph -> dom ( G o. `'inr ) = (inr " dom G ) )
39		df-fn 5378	. . . . 5 |- ( ( G o. `'inr ) Fn (inr " dom G ) <-> ( Fun ( G o. `'inr ) /\ dom ( G o. `'inr ) = (inr " dom G ) ) )
40	34, 38, 39	sylanbrc 421	. . . 4 |- ( ph -> ( G o. `'inr ) Fn (inr " dom G ) )
41		djuin 7397	. . . . 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 2835	. . . . . . 7 |- ( ph -> A e. _V )
45		f1odm 5641	. . . . . . . 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 2332	. . . . . 6 |- ( ph -> A e. dom inl )
48	47, 15	jctil 312	. . . . 5 |- ( ph -> ( Fun inl /\ A e. dom inl ) )
49		funfvima 5943	. . . . 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 5766	. . . 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 1282	. . 3 |- ( ph -> ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inl ` A ) ) = ( ( F o. `'inl ) ` (inl ` A ) ) )
53	2, 52	eqtrid 2283	. 2 |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( ( F o. `'inl ) ` (inl ` A ) ) )
54		f1ofn 5638	. . . 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 5637	. . . . . 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 5838	. . 3 |- ( ph -> (inl ` A ) e. ( { (/) } X. _V ) )
60		fvco2 5771	. . 3 |- ( ( `'inl Fn ( { (/) } X. _V ) /\ (inl ` A ) e. ( { (/) } X. _V ) ) -> ( ( F o. `'inl ) ` (inl ` A ) ) = ( F ` ( `'inl ` (inl ` A ) ) ) )
61	55, 59, 60	sylancr 418	. 2 |- ( ph -> ( ( F o. `'inl ) ` (inl ` A ) ) = ( F ` ( `'inl ` (inl ` A ) ) ) )
62		f1ocnvfv1 5976	. . . 4 |- ( (inl : _V -1-1-onto-> ( { (/) } X. _V ) /\ A e. _V ) -> ( `'inl ` (inl ` A ) ) = A )
63	6, 44, 62	sylancr 418	. . 3 |- ( ph -> ( `'inl ` (inl ` A ) ) = A )
64	63	fveq2d 5697	. 2 |- ( ph -> ( F ` ( `'inl ` (inl ` A ) ) ) = ( F ` A ) )
65	53, 61, 64	3eqtrd 2275	1 |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3708   <.cop 3711   X. cxp 4770   `'ccnv 4771   dom cdm 4772   "cima 4775   o. ccom 4776   Rel wrel 4777   Fun wfun 5369   Fn wfn 5370   -->wf 5371   -1-1-onto->wf1o 5374   ` cfv 5375   1oc1o 6673  inlcinl 7378  inrcinr 7379  casecdjucase 7416

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 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1st 6367  df-2nd 6368  df-1o 6680  df-inl 7380  df-inr 7381  df-case 7417

This theorem is referenced by:  omp1eomlem  7427  ctm  7442