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Theorem caserel 7146
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
StepHypRef Expression
1 df-case 7143 . 2 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
2 cocnvss 5191 . . . 4 (𝑅inl) ⊆ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl))
3 inlresf1 7120 . . . . . 6 (inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅 ⊔ dom 𝑆)
4 f1rn 5460 . . . . . 6 ((inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅 ⊔ dom 𝑆) → ran (inl ↾ dom 𝑅) ⊆ (dom 𝑅 ⊔ dom 𝑆))
53, 4ax-mp 5 . . . . 5 ran (inl ↾ dom 𝑅) ⊆ (dom 𝑅 ⊔ dom 𝑆)
6 resss 4966 . . . . . . 7 (𝑅 ↾ dom inl) ⊆ 𝑅
7 rnss 4892 . . . . . . 7 ((𝑅 ↾ dom inl) ⊆ 𝑅 → ran (𝑅 ↾ dom inl) ⊆ ran 𝑅)
86, 7ax-mp 5 . . . . . 6 ran (𝑅 ↾ dom inl) ⊆ ran 𝑅
9 ssun1 3322 . . . . . 6 ran 𝑅 ⊆ (ran 𝑅 ∪ ran 𝑆)
108, 9sstri 3188 . . . . 5 ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)
11 xpss12 4766 . . . . 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)))
125, 10, 11mp2an 426 . . . 4 (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
132, 12sstri 3188 . . 3 (𝑅inl) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
14 cocnvss 5191 . . . 4 (𝑆inr) ⊆ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr))
15 inrresf1 7121 . . . . . 6 (inr ↾ dom 𝑆):dom 𝑆1-1→(dom 𝑅 ⊔ dom 𝑆)
16 f1rn 5460 . . . . . 6 ((inr ↾ dom 𝑆):dom 𝑆1-1→(dom 𝑅 ⊔ dom 𝑆) → ran (inr ↾ dom 𝑆) ⊆ (dom 𝑅 ⊔ dom 𝑆))
1715, 16ax-mp 5 . . . . 5 ran (inr ↾ dom 𝑆) ⊆ (dom 𝑅 ⊔ dom 𝑆)
18 resss 4966 . . . . . . 7 (𝑆 ↾ dom inr) ⊆ 𝑆
19 rnss 4892 . . . . . . 7 ((𝑆 ↾ dom inr) ⊆ 𝑆 → ran (𝑆 ↾ dom inr) ⊆ ran 𝑆)
2018, 19ax-mp 5 . . . . . 6 ran (𝑆 ↾ dom inr) ⊆ ran 𝑆
21 ssun2 3323 . . . . . 6 ran 𝑆 ⊆ (ran 𝑅 ∪ ran 𝑆)
2220, 21sstri 3188 . . . . 5 ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)
23 xpss12 4766 . . . . 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)))
2417, 22, 23mp2an 426 . . . 4 (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
2514, 24sstri 3188 . . 3 (𝑆inr) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
2613, 25unssi 3334 . 2 ((𝑅inl) ∪ (𝑆inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
271, 26eqsstri 3211 1 case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
Colors of variables: wff set class
Syntax hints:  cun 3151  wss 3153   × cxp 4657  ccnv 4658  dom cdm 4659  ran crn 4660  cres 4661  ccom 4663  1-1wf1 5251  cdju 7096  inlcinl 7104  inrcinr 7105  casecdjucase 7142
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-dju 7097  df-inl 7106  df-inr 7107  df-case 7143
This theorem is referenced by:  casef  7147
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