ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caserel GIF version

Theorem caserel 7085
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 7082 . 2 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
2 cocnvss 5154 . . . 4 (𝑅inl) ⊆ (ran (inl ↾ dom 𝑅) × ran (𝑅 ↾ dom inl))
3 inlresf1 7059 . . . . . 6 (inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅 ⊔ dom 𝑆)
4 f1rn 5422 . . . . . 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 4931 . . . . . . 7 (𝑅 ↾ dom inl) ⊆ 𝑅
7 rnss 4857 . . . . . . 7 ((𝑅 ↾ dom inl) ⊆ 𝑅 → ran (𝑅 ↾ dom inl) ⊆ ran 𝑅)
86, 7ax-mp 5 . . . . . 6 ran (𝑅 ↾ dom inl) ⊆ ran 𝑅
9 ssun1 3298 . . . . . 6 ran 𝑅 ⊆ (ran 𝑅 ∪ ran 𝑆)
108, 9sstri 3164 . . . . 5 ran (𝑅 ↾ dom inl) ⊆ (ran 𝑅 ∪ ran 𝑆)
11 xpss12 4733 . . . . 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 3164 . . 3 (𝑅inl) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
14 cocnvss 5154 . . . 4 (𝑆inr) ⊆ (ran (inr ↾ dom 𝑆) × ran (𝑆 ↾ dom inr))
15 inrresf1 7060 . . . . . 6 (inr ↾ dom 𝑆):dom 𝑆1-1→(dom 𝑅 ⊔ dom 𝑆)
16 f1rn 5422 . . . . . 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 4931 . . . . . . 7 (𝑆 ↾ dom inr) ⊆ 𝑆
19 rnss 4857 . . . . . . 7 ((𝑆 ↾ dom inr) ⊆ 𝑆 → ran (𝑆 ↾ dom inr) ⊆ ran 𝑆)
2018, 19ax-mp 5 . . . . . 6 ran (𝑆 ↾ dom inr) ⊆ ran 𝑆
21 ssun2 3299 . . . . . 6 ran 𝑆 ⊆ (ran 𝑅 ∪ ran 𝑆)
2220, 21sstri 3164 . . . . 5 ran (𝑆 ↾ dom inr) ⊆ (ran 𝑅 ∪ ran 𝑆)
23 xpss12 4733 . . . . 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 3164 . . 3 (𝑆inr) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
2613, 25unssi 3310 . 2 ((𝑅inl) ∪ (𝑆inr)) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
271, 26eqsstri 3187 1 case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
Colors of variables: wff set class
Syntax hints:  cun 3127  wss 3129   × cxp 4624  ccnv 4625  dom cdm 4626  ran crn 4627  cres 4628  ccom 4630  1-1wf1 5213  cdju 7035  inlcinl 7043  inrcinr 7044  casecdjucase 7081
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 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-dju 7036  df-inl 7045  df-inr 7046  df-case 7082
This theorem is referenced by:  casef  7086
  Copyright terms: Public domain W3C validator