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Theorem caserel 7132
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 7129 . 2 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
2 cocnvss 5179 . . . 4 (𝑅inl) ⊆ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl))
3 inlresf1 7106 . . . . . 6 (inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅 ⊔ dom 𝑆)
4 f1rn 5448 . . . . . 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 4956 . . . . . . 7 (𝑅 ↾ dom inl) ⊆ 𝑅
7 rnss 4882 . . . . . . 7 ((𝑅 ↾ dom inl) ⊆ 𝑅 → ran (𝑅 ↾ dom inl) ⊆ ran 𝑅)
86, 7ax-mp 5 . . . . . 6 ran (𝑅 ↾ dom inl) ⊆ ran 𝑅
9 ssun1 3318 . . . . . 6 ran 𝑅 ⊆ (ran 𝑅 ∪ ran 𝑆)
108, 9sstri 3184 . . . . 5 ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)
11 xpss12 4758 . . . . 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 3184 . . 3 (𝑅inl) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
14 cocnvss 5179 . . . 4 (𝑆inr) ⊆ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr))
15 inrresf1 7107 . . . . . 6 (inr ↾ dom 𝑆):dom 𝑆1-1→(dom 𝑅 ⊔ dom 𝑆)
16 f1rn 5448 . . . . . 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 4956 . . . . . . 7 (𝑆 ↾ dom inr) ⊆ 𝑆
19 rnss 4882 . . . . . . 7 ((𝑆 ↾ dom inr) ⊆ 𝑆 → ran (𝑆 ↾ dom inr) ⊆ ran 𝑆)
2018, 19ax-mp 5 . . . . . 6 ran (𝑆 ↾ dom inr) ⊆ ran 𝑆
21 ssun2 3319 . . . . . 6 ran 𝑆 ⊆ (ran 𝑅 ∪ ran 𝑆)
2220, 21sstri 3184 . . . . 5 ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)
23 xpss12 4758 . . . . 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 3184 . . 3 (𝑆inr) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
2613, 25unssi 3330 . 2 ((𝑅inl) ∪ (𝑆inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
271, 26eqsstri 3207 1 case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
Colors of variables: wff set class
Syntax hints:  cun 3147  wss 3149   × cxp 4649  ccnv 4650  dom cdm 4651  ran crn 4652  cres 4653  ccom 4655  1-1wf1 5239  cdju 7082  inlcinl 7090  inrcinr 7091  casecdjucase 7128
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2758  df-sbc 2982  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-id 4318  df-iord 4391  df-on 4393  df-suc 4396  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-1st 6180  df-2nd 6181  df-1o 6456  df-dju 7083  df-inl 7092  df-inr 7093  df-case 7129
This theorem is referenced by:  casef  7133
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