Theorem caserel 7076

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 7073	. 2 |- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
2		cocnvss 5146	. . . 4 |- ( R o. `'inl ) C_ ( ran (inl |` dom R ) X. ran ( R |` dom inl ) )
3		inlresf1 7050	. . . . . 6 |- (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ dom S )
4		f1rn 5414	. . . . . 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 4924	. . . . . . 7 |- ( R |` dom inl ) C_ R
7		rnss 4850	. . . . . . 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 3296	. . . . . 6 |- ran R C_ ( ran R u. ran S )
10	8, 9	sstri 3162	. . . . 5 |- ran ( R |` dom inl ) C_ ( ran R u. ran S )
11		xpss12 4727	. . . . 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 3162	. . 3 |- ( R o. `'inl ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
14		cocnvss 5146	. . . 4 |- ( S o. `'inr ) C_ ( ran (inr |` dom S ) X. ran ( S |` dom inr ) )
15		inrresf1 7051	. . . . . 6 |- (inr |` dom S ) : dom S -1-1-> ( dom R ⊔ dom S )
16		f1rn 5414	. . . . . 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 4924	. . . . . . 7 |- ( S |` dom inr ) C_ S
19		rnss 4850	. . . . . . 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 3297	. . . . . 6 |- ran S C_ ( ran R u. ran S )
22	20, 21	sstri 3162	. . . . 5 |- ran ( S |` dom inr ) C_ ( ran R u. ran S )
23		xpss12 4727	. . . . 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 3162	. . 3 |- ( S o. `'inr ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
26	13, 25	unssi 3308	. 2 |- ( ( R o. `'inl ) u. ( S o. `'inr ) ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
27	1, 26	eqsstri 3185	1 |- case ( R , S ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )

Syntax hints:   u. cun 3125   C_ wss 3127   X. cxp 4618   `'ccnv 4619   dom cdm 4620   ran crn 4621   |` cres 4622   o. ccom 4624   -1-1->wf1 5205   ⊔ cdju 7026  inlcinl 7034  inrcinr 7035  casecdjucase 7072

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-dju 7027  df-inl 7036  df-inr 7037  df-case 7073

This theorem is referenced by:  casef  7077