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Theorem caserel 7088
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 7085 . 2 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
2 cocnvss 5156 . . . 4 (𝑅inl) ⊆ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl))
3 inlresf1 7062 . . . . . 6 (inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅 ⊔ dom 𝑆)
4 f1rn 5424 . . . . . 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 4933 . . . . . . 7 (𝑅 ↾ dom inl) ⊆ 𝑅
7 rnss 4859 . . . . . . 7 ((𝑅 ↾ dom inl) ⊆ 𝑅 → ran (𝑅 ↾ dom inl) ⊆ ran 𝑅)
86, 7ax-mp 5 . . . . . 6 ran (𝑅 ↾ dom inl) ⊆ ran 𝑅
9 ssun1 3300 . . . . . 6 ran 𝑅 ⊆ (ran 𝑅 ∪ ran 𝑆)
108, 9sstri 3166 . . . . 5 ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)
11 xpss12 4735 . . . . 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 3166 . . 3 (𝑅inl) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
14 cocnvss 5156 . . . 4 (𝑆inr) ⊆ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr))
15 inrresf1 7063 . . . . . 6 (inr ↾ dom 𝑆):dom 𝑆1-1→(dom 𝑅 ⊔ dom 𝑆)
16 f1rn 5424 . . . . . 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 4933 . . . . . . 7 (𝑆 ↾ dom inr) ⊆ 𝑆
19 rnss 4859 . . . . . . 7 ((𝑆 ↾ dom inr) ⊆ 𝑆 → ran (𝑆 ↾ dom inr) ⊆ ran 𝑆)
2018, 19ax-mp 5 . . . . . 6 ran (𝑆 ↾ dom inr) ⊆ ran 𝑆
21 ssun2 3301 . . . . . 6 ran 𝑆 ⊆ (ran 𝑅 ∪ ran 𝑆)
2220, 21sstri 3166 . . . . 5 ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)
23 xpss12 4735 . . . . 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 3166 . . 3 (𝑆inr) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
2613, 25unssi 3312 . 2 ((𝑅inl) ∪ (𝑆inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
271, 26eqsstri 3189 1 case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
Colors of variables: wff set class
Syntax hints:  cun 3129  wss 3131   × cxp 4626  ccnv 4627  dom cdm 4628  ran crn 4629  cres 4630  ccom 4632  1-1wf1 5215  cdju 7038  inlcinl 7046  inrcinr 7047  casecdjucase 7084
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-dju 7039  df-inl 7048  df-inr 7049  df-case 7085
This theorem is referenced by:  casef  7089
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