Theorem caserel 7146

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 7143	. 2 |- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
2		cocnvss 5191	. . . 4 |- ( R o. `'inl ) C_ ( ran (inl |` dom R ) X. ran ( R |` dom inl ) )
3		inlresf1 7120	. . . . . 6 |- (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ dom S )
4		f1rn 5460	. . . . . 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 5	. . . . 5 |- ran (inl |` dom R ) C_ ( dom R ⊔ dom S )
6		resss 4966	. . . . . . 7 |- ( R |` dom inl ) C_ R
7		rnss 4892	. . . . . . 7 |- ( ( R |` dom inl ) C_ R -> ran ( R |` dom inl ) C_ ran R )
8	6, 7	ax-mp 5	. . . . . 6 |- ran ( R |` dom inl ) C_ ran R
9		ssun1 3322	. . . . . 6 |- ran R C_ ( ran R u. ran S )
10	8, 9	sstri 3188	. . . . 5 |- ran ( R |` dom inl ) C_ ( ran R u. ran S )
11		xpss12 4766	. . . . 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 426	. . . 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 3188	. . 3 |- ( R o. `'inl ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
14		cocnvss 5191	. . . 4 |- ( S o. `'inr ) C_ ( ran (inr |` dom S ) X. ran ( S |` dom inr ) )
15		inrresf1 7121	. . . . . 6 |- (inr |` dom S ) : dom S -1-1-> ( dom R ⊔ dom S )
16		f1rn 5460	. . . . . 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 5	. . . . 5 |- ran (inr |` dom S ) C_ ( dom R ⊔ dom S )
18		resss 4966	. . . . . . 7 |- ( S |` dom inr ) C_ S
19		rnss 4892	. . . . . . 7 |- ( ( S |` dom inr ) C_ S -> ran ( S |` dom inr ) C_ ran S )
20	18, 19	ax-mp 5	. . . . . 6 |- ran ( S |` dom inr ) C_ ran S
21		ssun2 3323	. . . . . 6 |- ran S C_ ( ran R u. ran S )
22	20, 21	sstri 3188	. . . . 5 |- ran ( S |` dom inr ) C_ ( ran R u. ran S )
23		xpss12 4766	. . . . 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 426	. . . 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 3188	. . 3 |- ( S o. `'inr ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
26	13, 25	unssi 3334	. 2 |- ( ( R o. `'inl ) u. ( S o. `'inr ) ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
27	1, 26	eqsstri 3211	1 |- case ( R , S ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )

Syntax hints:   u. cun 3151   C_ wss 3153   X. cxp 4657   `'ccnv 4658   dom cdm 4659   ran crn 4660   |` cres 4661   o. ccom 4663   -1-1->wf1 5251   ⊔ cdju 7096  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-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  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-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-dju 7097  df-inl 7106  df-inr 7107  df-case 7143

This theorem is referenced by:  casef  7147