Theorem caserel 7052

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 7049	. 2 ⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
2		cocnvss 5129	. . . 4 ⊢ (𝑅 ∘ ◡inl) ⊆ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl))
3		inlresf1 7026	. . . . . 6 ⊢ (inl ↾ dom 𝑅):dom 𝑅–1-1→(dom 𝑅 ⊔ dom 𝑆)
4		f1rn 5394	. . . . . 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 4908	. . . . . . 7 ⊢ (𝑅 ↾ dom inl) ⊆ 𝑅
7		rnss 4834	. . . . . . 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 3285	. . . . . 6 ⊢ ran 𝑅 ⊆ (ran 𝑅 ∪ ran 𝑆)
10	8, 9	sstri 3151	. . . . 5 ⊢ ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)
11		xpss12 4711	. . . . 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 423	. . . 4 ⊢ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
13	2, 12	sstri 3151	. . 3 ⊢ (𝑅 ∘ ◡inl) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
14		cocnvss 5129	. . . 4 ⊢ (𝑆 ∘ ◡inr) ⊆ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr))
15		inrresf1 7027	. . . . . 6 ⊢ (inr ↾ dom 𝑆):dom 𝑆–1-1→(dom 𝑅 ⊔ dom 𝑆)
16		f1rn 5394	. . . . . 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 4908	. . . . . . 7 ⊢ (𝑆 ↾ dom inr) ⊆ 𝑆
19		rnss 4834	. . . . . . 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 3286	. . . . . 6 ⊢ ran 𝑆 ⊆ (ran 𝑅 ∪ ran 𝑆)
22	20, 21	sstri 3151	. . . . 5 ⊢ ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)
23		xpss12 4711	. . . . 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 423	. . . 4 ⊢ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
25	14, 24	sstri 3151	. . 3 ⊢ (𝑆 ∘ ◡inr) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
26	13, 25	unssi 3297	. 2 ⊢ ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
27	1, 26	eqsstri 3174	1 ⊢ case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))

Syntax hints:   ∪ cun 3114   ⊆ wss 3116   × cxp 4602  ^◡ccnv 4603  dom cdm 4604  ran crn 4605   ↾ cres 4606   ∘ ccom 4608  –1-1→wf1 5185   ⊔ cdju 7002  inlcinl 7010  inrcinr 7011  casecdjucase 7048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013  df-case 7049

This theorem is referenced by:  casef  7053