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Theorem caserel 6972
 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 6969 . 2 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
2 cocnvss 5064 . . . 4 (𝑅inl) ⊆ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl))
3 inlresf1 6946 . . . . . 6 (inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅 ⊔ dom 𝑆)
4 f1rn 5329 . . . . . 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 4843 . . . . . . 7 (𝑅 ↾ dom inl) ⊆ 𝑅
7 rnss 4769 . . . . . . 7 ((𝑅 ↾ dom inl) ⊆ 𝑅 → ran (𝑅 ↾ dom inl) ⊆ ran 𝑅)
86, 7ax-mp 5 . . . . . 6 ran (𝑅 ↾ dom inl) ⊆ ran 𝑅
9 ssun1 3239 . . . . . 6 ran 𝑅 ⊆ (ran 𝑅 ∪ ran 𝑆)
108, 9sstri 3106 . . . . 5 ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)
11 xpss12 4646 . . . . 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 422 . . . 4 (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
132, 12sstri 3106 . . 3 (𝑅inl) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
14 cocnvss 5064 . . . 4 (𝑆inr) ⊆ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr))
15 inrresf1 6947 . . . . . 6 (inr ↾ dom 𝑆):dom 𝑆1-1→(dom 𝑅 ⊔ dom 𝑆)
16 f1rn 5329 . . . . . 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 4843 . . . . . . 7 (𝑆 ↾ dom inr) ⊆ 𝑆
19 rnss 4769 . . . . . . 7 ((𝑆 ↾ dom inr) ⊆ 𝑆 → ran (𝑆 ↾ dom inr) ⊆ ran 𝑆)
2018, 19ax-mp 5 . . . . . 6 ran (𝑆 ↾ dom inr) ⊆ ran 𝑆
21 ssun2 3240 . . . . . 6 ran 𝑆 ⊆ (ran 𝑅 ∪ ran 𝑆)
2220, 21sstri 3106 . . . . 5 ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)
23 xpss12 4646 . . . . 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 422 . . . 4 (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
2514, 24sstri 3106 . . 3 (𝑆inr) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
2613, 25unssi 3251 . 2 ((𝑅inl) ∪ (𝑆inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
271, 26eqsstri 3129 1 case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
 Colors of variables: wff set class Syntax hints:   ∪ cun 3069   ⊆ wss 3071   × cxp 4537  ◡ccnv 4538  dom cdm 4539  ran crn 4540   ↾ cres 4541   ∘ ccom 4543  –1-1→wf1 5120   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931  casecdjucase 6968 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933  df-case 6969 This theorem is referenced by:  casef  6973
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