Theorem caserel 7277

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 7274	. 2 ⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
2		cocnvss 5260	. . . 4 ⊢ (𝑅 ∘ ◡inl) ⊆ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl))
3		inlresf1 7251	. . . . . 6 ⊢ (inl ↾ dom 𝑅):dom 𝑅–1-1→(dom 𝑅 ⊔ dom 𝑆)
4		f1rn 5540	. . . . . 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 5035	. . . . . . 7 ⊢ (𝑅 ↾ dom inl) ⊆ 𝑅
7		rnss 4960	. . . . . . 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 3368	. . . . . 6 ⊢ ran 𝑅 ⊆ (ran 𝑅 ∪ ran 𝑆)
10	8, 9	sstri 3234	. . . . 5 ⊢ ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)
11		xpss12 4831	. . . . 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 3234	. . 3 ⊢ (𝑅 ∘ ◡inl) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
14		cocnvss 5260	. . . 4 ⊢ (𝑆 ∘ ◡inr) ⊆ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr))
15		inrresf1 7252	. . . . . 6 ⊢ (inr ↾ dom 𝑆):dom 𝑆–1-1→(dom 𝑅 ⊔ dom 𝑆)
16		f1rn 5540	. . . . . 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 5035	. . . . . . 7 ⊢ (𝑆 ↾ dom inr) ⊆ 𝑆
19		rnss 4960	. . . . . . 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 3369	. . . . . 6 ⊢ ran 𝑆 ⊆ (ran 𝑅 ∪ ran 𝑆)
22	20, 21	sstri 3234	. . . . 5 ⊢ ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)
23		xpss12 4831	. . . . 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 3234	. . 3 ⊢ (𝑆 ∘ ◡inr) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
26	13, 25	unssi 3380	. 2 ⊢ ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
27	1, 26	eqsstri 3257	1 ⊢ case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))

Syntax hints:   ∪ cun 3196   ⊆ wss 3198   × cxp 4721  ^◡ccnv 4722  dom cdm 4723  ran crn 4724   ↾ cres 4725   ∘ ccom 4727  –1-1→wf1 5321   ⊔ cdju 7227  inlcinl 7235  inrcinr 7236  casecdjucase 7273

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1st 6298  df-2nd 6299  df-1o 6577  df-dju 7228  df-inl 7237  df-inr 7238  df-case 7274

This theorem is referenced by:  casef  7278