Theorem caserel 7153

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 7150	. 2 ⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
2		cocnvss 5195	. . . 4 ⊢ (𝑅 ∘ ◡inl) ⊆ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl))
3		inlresf1 7127	. . . . . 6 ⊢ (inl ↾ dom 𝑅):dom 𝑅–1-1→(dom 𝑅 ⊔ dom 𝑆)
4		f1rn 5464	. . . . . 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 4970	. . . . . . 7 ⊢ (𝑅 ↾ dom inl) ⊆ 𝑅
7		rnss 4896	. . . . . . 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 3326	. . . . . 6 ⊢ ran 𝑅 ⊆ (ran 𝑅 ∪ ran 𝑆)
10	8, 9	sstri 3192	. . . . 5 ⊢ ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)
11		xpss12 4770	. . . . 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 3192	. . 3 ⊢ (𝑅 ∘ ◡inl) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
14		cocnvss 5195	. . . 4 ⊢ (𝑆 ∘ ◡inr) ⊆ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr))
15		inrresf1 7128	. . . . . 6 ⊢ (inr ↾ dom 𝑆):dom 𝑆–1-1→(dom 𝑅 ⊔ dom 𝑆)
16		f1rn 5464	. . . . . 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 4970	. . . . . . 7 ⊢ (𝑆 ↾ dom inr) ⊆ 𝑆
19		rnss 4896	. . . . . . 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 3327	. . . . . 6 ⊢ ran 𝑆 ⊆ (ran 𝑅 ∪ ran 𝑆)
22	20, 21	sstri 3192	. . . . 5 ⊢ ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)
23		xpss12 4770	. . . . 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 3192	. . 3 ⊢ (𝑆 ∘ ◡inr) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
26	13, 25	unssi 3338	. 2 ⊢ ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
27	1, 26	eqsstri 3215	1 ⊢ case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))

Syntax hints:   ∪ cun 3155   ⊆ wss 3157   × cxp 4661  ^◡ccnv 4662  dom cdm 4663  ran crn 4664   ↾ cres 4665   ∘ ccom 4667  –1-1→wf1 5255   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112  casecdjucase 7149

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114  df-case 7150

This theorem is referenced by:  casef  7154