Theorem caserel 7329

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) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))

Proof of Theorem caserel

Step	Hyp	Ref	Expression
1		df-case 7326	. 2 ⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
2		cocnvss 5269	. . . 4 ⊢ (𝑅 ∘ ◡inl) ⊆ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl))
3		inlresf1 7303	. . . . . 6 ⊢ (inl ↾ dom 𝑅):dom 𝑅–1-1→(dom 𝑅 ⊔ dom 𝑆)
4		f1rn 5552	. . . . . 6 ⊢ ((inl ↾ dom 𝑅):dom 𝑅–1-1→(dom 𝑅 ⊔ dom 𝑆) → ran (inl ↾ dom 𝑅) ⊆ (dom 𝑅 ⊔ dom 𝑆))
5	3, 4	ax-mp 5	. . . . 5 ⊢ ran (inl ↾ dom 𝑅) ⊆ (dom 𝑅 ⊔ dom 𝑆)
6		resss 5043	. . . . . . 7 ⊢ (𝑅 ↾ dom inl) ⊆ 𝑅
7		rnss 4968	. . . . . . 7 ⊢ ((𝑅 ↾ dom inl) ⊆ 𝑅 → ran (𝑅 ↾ dom inl) ⊆ ran 𝑅)
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ran (𝑅 ↾ dom inl) ⊆ ran 𝑅
9		ssun1 3372	. . . . . 6 ⊢ ran 𝑅 ⊆ (ran 𝑅 ∪ ran 𝑆)
10	8, 9	sstri 3237	. . . . 5 ⊢ ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)
11		xpss12 4839	. . . . 5 ⊢ ((ran (inl ↾ dom 𝑅) ⊆ (dom 𝑅 ⊔ dom 𝑆) ∧ ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)) → (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆)))
12	5, 10, 11	mp2an 426	. . . 4 ⊢ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
13	2, 12	sstri 3237	. . 3 ⊢ (𝑅 ∘ ◡inl) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
14		cocnvss 5269	. . . 4 ⊢ (𝑆 ∘ ◡inr) ⊆ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr))
15		inrresf1 7304	. . . . . 6 ⊢ (inr ↾ dom 𝑆):dom 𝑆–1-1→(dom 𝑅 ⊔ dom 𝑆)
16		f1rn 5552	. . . . . 6 ⊢ ((inr ↾ dom 𝑆):dom 𝑆–1-1→(dom 𝑅 ⊔ dom 𝑆) → ran (inr ↾ dom 𝑆) ⊆ (dom 𝑅 ⊔ dom 𝑆))
17	15, 16	ax-mp 5	. . . . 5 ⊢ ran (inr ↾ dom 𝑆) ⊆ (dom 𝑅 ⊔ dom 𝑆)
18		resss 5043	. . . . . . 7 ⊢ (𝑆 ↾ dom inr) ⊆ 𝑆
19		rnss 4968	. . . . . . 7 ⊢ ((𝑆 ↾ dom inr) ⊆ 𝑆 → ran (𝑆 ↾ dom inr) ⊆ ran 𝑆)
20	18, 19	ax-mp 5	. . . . . 6 ⊢ ran (𝑆 ↾ dom inr) ⊆ ran 𝑆
21		ssun2 3373	. . . . . 6 ⊢ ran 𝑆 ⊆ (ran 𝑅 ∪ ran 𝑆)
22	20, 21	sstri 3237	. . . . 5 ⊢ ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)
23		xpss12 4839	. . . . 5 ⊢ ((ran (inr ↾ dom 𝑆) ⊆ (dom 𝑅 ⊔ dom 𝑆) ∧ ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)) → (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆)))
24	17, 22, 23	mp2an 426	. . . 4 ⊢ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
25	14, 24	sstri 3237	. . 3 ⊢ (𝑆 ∘ ◡inr) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
26	13, 25	unssi 3384	. 2 ⊢ ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
27	1, 26	eqsstri 3260	1 ⊢ case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))

Syntax hints:   ∪ cun 3199   ⊆ wss 3201   × cxp 4729  ^◡ccnv 4730  dom cdm 4731  ran crn 4732   ↾ cres 4733   ∘ ccom 4735  –1-1→wf1 5330   ⊔ cdju 7279  inlcinl 7287  inrcinr 7288  casecdjucase 7325

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-dju 7280  df-inl 7289  df-inr 7290  df-case 7326

This theorem is referenced by:  casef  7330