Theorem caserel 6834

Description: The "case" construction of two relations is a relation, with bounds on its domain and codomain. Typically, the "case" construction is used when both relations have a common codomain. (Contributed by BJ, 10-Jul-2022.)

Assertion

Ref	Expression
caserel	|- case ( R , S ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )

Proof of Theorem caserel

Step	Hyp	Ref	Expression
1		df-case 6831	. 2 |- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
2		cocnvss 4971	. . . 4 |- ( R o. `'inl ) C_ ( ran (inl |` dom R ) X. ran ( R |` dom inl ) )
3		inlresf1 6809	. . . . . 6 |- (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ dom S )
4		f1rn 5232	. . . . . 6 |- ( (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ dom S ) -> ran (inl |` dom R ) C_ ( dom R ⊔ dom S ) )
5	3, 4	ax-mp 7	. . . . 5 |- ran (inl |` dom R ) C_ ( dom R ⊔ dom S )
6		resss 4752	. . . . . . 7 |- ( R |` dom inl ) C_ R
7		rnss 4680	. . . . . . 7 |- ( ( R |` dom inl ) C_ R -> ran ( R |` dom inl ) C_ ran R )
8	6, 7	ax-mp 7	. . . . . 6 |- ran ( R |` dom inl ) C_ ran R
9		ssun1 3166	. . . . . 6 |- ran R C_ ( ran R u. ran S )
10	8, 9	sstri 3037	. . . . 5 |- ran ( R |` dom inl ) C_ ( ran R u. ran S )
11		xpss12 4560	. . . . 5 |- ( ( ran (inl |` dom R ) C_ ( dom R ⊔ dom S ) /\ ran ( R |` dom inl ) C_ ( ran R u. ran S ) ) -> ( ran (inl |` dom R ) X. ran ( R |` dom inl ) ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) ) )
12	5, 10, 11	mp2an 418	. . . 4 |- ( ran (inl |` dom R ) X. ran ( R |` dom inl ) ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
13	2, 12	sstri 3037	. . 3 |- ( R o. `'inl ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
14		cocnvss 4971	. . . 4 |- ( S o. `'inr ) C_ ( ran (inr |` dom S ) X. ran ( S |` dom inr ) )
15		inrresf1 6810	. . . . . 6 |- (inr |` dom S ) : dom S -1-1-> ( dom R ⊔ dom S )
16		f1rn 5232	. . . . . 6 |- ( (inr |` dom S ) : dom S -1-1-> ( dom R ⊔ dom S ) -> ran (inr |` dom S ) C_ ( dom R ⊔ dom S ) )
17	15, 16	ax-mp 7	. . . . 5 |- ran (inr |` dom S ) C_ ( dom R ⊔ dom S )
18		resss 4752	. . . . . . 7 |- ( S |` dom inr ) C_ S
19		rnss 4680	. . . . . . 7 |- ( ( S |` dom inr ) C_ S -> ran ( S |` dom inr ) C_ ran S )
20	18, 19	ax-mp 7	. . . . . 6 |- ran ( S |` dom inr ) C_ ran S
21		ssun2 3167	. . . . . 6 |- ran S C_ ( ran R u. ran S )
22	20, 21	sstri 3037	. . . . 5 |- ran ( S |` dom inr ) C_ ( ran R u. ran S )
23		xpss12 4560	. . . . 5 |- ( ( ran (inr |` dom S ) C_ ( dom R ⊔ dom S ) /\ ran ( S |` dom inr ) C_ ( ran R u. ran S ) ) -> ( ran (inr |` dom S ) X. ran ( S |` dom inr ) ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) ) )
24	17, 22, 23	mp2an 418	. . . 4 |- ( ran (inr |` dom S ) X. ran ( S |` dom inr ) ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
25	14, 24	sstri 3037	. . 3 |- ( S o. `'inr ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
26	13, 25	unssi 3178	. 2 |- ( ( R o. `'inl ) u. ( S o. `'inr ) ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
27	1, 26	eqsstri 3059	1 |- case ( R , S ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )

Syntax hints:   u. cun 3000   C_ wss 3002   X. cxp 4452   `'ccnv 4453   dom cdm 4454   ran crn 4455   |` cres 4456   o. ccom 4458   -1-1->wf1 5027   ⊔ cdju 6786  inlcinl 6793  inrcinr 6794  casecdjucase 6830

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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-iord 4204  df-on 4206  df-suc 4209  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-1st 5927  df-2nd 5928  df-1o 6197  df-dju 6787  df-inl 6795  df-inr 6796  df-case 6831

This theorem is referenced by:  casef  6835