Theorem caserel 7393

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 7390	. 2 |- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
2		cocnvss 5295	. . . 4 |- ( R o. `'inl ) C_ ( ran (inl |` dom R ) X. ran ( R |` dom inl ) )
3		inlresf1 7367	. . . . . 6 |- (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ dom S )
4		f1rn 5581	. . . . . 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 5069	. . . . . . 7 |- ( R |` dom inl ) C_ R
7		rnss 4994	. . . . . . 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 3386	. . . . . 6 |- ran R C_ ( ran R u. ran S )
10	8, 9	sstri 3251	. . . . 5 |- ran ( R |` dom inl ) C_ ( ran R u. ran S )
11		xpss12 4864	. . . . 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 3251	. . 3 |- ( R o. `'inl ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
14		cocnvss 5295	. . . 4 |- ( S o. `'inr ) C_ ( ran (inr |` dom S ) X. ran ( S |` dom inr ) )
15		inrresf1 7368	. . . . . 6 |- (inr |` dom S ) : dom S -1-1-> ( dom R ⊔ dom S )
16		f1rn 5581	. . . . . 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 5069	. . . . . . 7 |- ( S |` dom inr ) C_ S
19		rnss 4994	. . . . . . 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 3387	. . . . . 6 |- ran S C_ ( ran R u. ran S )
22	20, 21	sstri 3251	. . . . 5 |- ran ( S |` dom inr ) C_ ( ran R u. ran S )
23		xpss12 4864	. . . . 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 3251	. . 3 |- ( S o. `'inr ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
26	13, 25	unssi 3398	. 2 |- ( ( R o. `'inl ) u. ( S o. `'inr ) ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
27	1, 26	eqsstri 3274	1 |- case ( R , S ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )

Syntax hints:   u. cun 3212   C_ wss 3214   X. cxp 4754   `'ccnv 4755   dom cdm 4756   ran crn 4757   |` cres 4758   o. ccom 4760   -1-1->wf1 5356   ⊔ cdju 7343  inlcinl 7351  inrcinr 7352  casecdjucase 7389

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-14 2208  ax-ext 2216  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-iord 4493  df-on 4495  df-suc 4498  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-1st 6349  df-2nd 6350  df-1o 6662  df-dju 7344  df-inl 7353  df-inr 7354  df-case 7390

This theorem is referenced by:  casef  7394