Theorem caseinl 7384

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 7377	. . . 4 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
2	1	fveq1i 5673	. . 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 5455	. . . . . . 7 |- ( F Fn B -> Fun F )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> Fun F )
6		djulf1o 7351	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
7		f1ocnv 5629	. . . . . . . 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 5618	. . . . . . 7 |- ( `'inl : ( { (/) } X. _V ) -1-1-onto-> _V -> Fun `'inl )
10	8, 9	ax-mp 5	. . . . . 6 |- Fun `'inl
11		funco 5394	. . . . . 6 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
12	5, 10, 11	sylancl 413	. . . . 5 |- ( ph -> Fun ( F o. `'inl ) )
13		dmco 5273	. . . . . 6 |- dom ( F o. `'inl ) = ( `' `'inl " dom F )
14		df-inl 7340	. . . . . . . . . . 11 |- inl = ( x e. _V |-> <. (/) , x >. )
15	14	funmpt2 5393	. . . . . . . . . 10 |- Fun inl
16		funrel 5371	. . . . . . . . . 10 |- ( Fun inl -> Rel inl )
17	15, 16	ax-mp 5	. . . . . . . . 9 |- Rel inl
18		dfrel2 5215	. . . . . . . . 9 |- ( Rel inl <-> `' `'inl = inl )
19	17, 18	mpbi 145	. . . . . . . 8 |- `' `'inl = inl
20	19	a1i 9	. . . . . . 7 |- ( ph -> `' `'inl = inl )
21		fndm 5457	. . . . . . . 8 |- ( F Fn B -> dom F = B )
22	3, 21	syl 14	. . . . . . 7 |- ( ph -> dom F = B )
23	20, 22	imaeq12d 5104	. . . . . 6 |- ( ph -> ( `' `'inl " dom F ) = (inl " B ) )
24	13, 23	eqtrid 2279	. . . . 5 |- ( ph -> dom ( F o. `'inl ) = (inl " B ) )
25		df-fn 5357	. . . . 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 7352	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
29		f1ocnv 5629	. . . . . . . 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 5618	. . . . . . 7 |- ( `'inr : ( { 1o } X. _V ) -1-1-onto-> _V -> Fun `'inr )
32	30, 31	ax-mp 5	. . . . . 6 |- Fun `'inr
33		funco 5394	. . . . . 6 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
34	27, 32, 33	sylancl 413	. . . . 5 |- ( ph -> Fun ( G o. `'inr ) )
35		dmco 5273	. . . . . . 7 |- dom ( G o. `'inr ) = ( `' `'inr " dom G )
36		imacnvcnv 5229	. . . . . . 7 |- ( `' `'inr " dom G ) = (inr " dom G )
37	35, 36	eqtri 2255	. . . . . 6 |- dom ( G o. `'inr ) = (inr " dom G )
38	37	a1i 9	. . . . 5 |- ( ph -> dom ( G o. `'inr ) = (inr " dom G ) )
39		df-fn 5357	. . . . 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 7357	. . . . 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 2829	. . . . . . 7 |- ( ph -> A e. _V )
45		f1odm 5620	. . . . . . . 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 2328	. . . . . 6 |- ( ph -> A e. dom inl )
48	47, 15	jctil 312	. . . . 5 |- ( ph -> ( Fun inl /\ A e. dom inl ) )
49		funfvima 5920	. . . . 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 5745	. . . 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 1278	. . 3 |- ( ph -> ( ( ( F o. `'inl ) u. ( G o. `'inr ) ) ` (inl ` A ) ) = ( ( F o. `'inl ) ` (inl ` A ) ) )
53	2, 52	eqtrid 2279	. 2 |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( ( F o. `'inl ) ` (inl ` A ) ) )
54		f1ofn 5617	. . . 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 5616	. . . . . 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 5815	. . 3 |- ( ph -> (inl ` A ) e. ( { (/) } X. _V ) )
60		fvco2 5748	. . 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 5952	. . . 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 5676	. 2 |- ( ph -> ( F ` ( `'inl ` (inl ` A ) ) ) = ( F ` A ) )
65	53, 61, 64	3eqtrd 2271	1 |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   u. cun 3211   i^i cin 3212   (/)c0 3510   {csn 3691   <.cop 3694   X. cxp 4749   `'ccnv 4750   dom cdm 4751   "cima 4754   o. ccom 4755   Rel wrel 4756   Fun wfun 5348   Fn wfn 5349   -->wf 5350   -1-1-onto->wf1o 5353   ` cfv 5354   1oc1o 6642  inlcinl 7338  inrcinr 7339  casecdjucase 7376

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-1st 6336  df-2nd 6337  df-1o 6649  df-inl 7340  df-inr 7341  df-case 7377

This theorem is referenced by:  omp1eomlem  7387  ctm  7402