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Theorem djuinj 6984
Description: The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
djuinj.r (𝜑 → Fun 𝑅)
djuinj.s (𝜑 → Fun 𝑆)
djuinj.disj (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)
Assertion
Ref Expression
djuinj (𝜑 → Fun (𝑅d 𝑆))

Proof of Theorem djuinj
StepHypRef Expression
1 inlresf1 6939 . . . . . . 7 (inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅𝐴)
2 f1fun 5326 . . . . . . 7 ((inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅𝐴) → Fun (inl ↾ dom 𝑅))
31, 2ax-mp 5 . . . . . 6 Fun (inl ↾ dom 𝑅)
4 funcnvcnv 5177 . . . . . 6 (Fun (inl ↾ dom 𝑅) → Fun (inl ↾ dom 𝑅))
53, 4ax-mp 5 . . . . 5 Fun (inl ↾ dom 𝑅)
6 djuinj.r . . . . 5 (𝜑 → Fun 𝑅)
7 funco 5158 . . . . 5 ((Fun (inl ↾ dom 𝑅) ∧ Fun 𝑅) → Fun ((inl ↾ dom 𝑅) ∘ 𝑅))
85, 6, 7sylancr 410 . . . 4 (𝜑 → Fun ((inl ↾ dom 𝑅) ∘ 𝑅))
9 cnvco 4719 . . . . 5 (𝑅(inl ↾ dom 𝑅)) = ((inl ↾ dom 𝑅) ∘ 𝑅)
109funeqi 5139 . . . 4 (Fun (𝑅(inl ↾ dom 𝑅)) ↔ Fun ((inl ↾ dom 𝑅) ∘ 𝑅))
118, 10sylibr 133 . . 3 (𝜑 → Fun (𝑅(inl ↾ dom 𝑅)))
12 inrresf1 6940 . . . . . . 7 (inr ↾ dom 𝑆):dom 𝑆1-1→(𝐴 ⊔ dom 𝑆)
13 f1fun 5326 . . . . . . 7 ((inr ↾ dom 𝑆):dom 𝑆1-1→(𝐴 ⊔ dom 𝑆) → Fun (inr ↾ dom 𝑆))
1412, 13ax-mp 5 . . . . . 6 Fun (inr ↾ dom 𝑆)
15 funcnvcnv 5177 . . . . . 6 (Fun (inr ↾ dom 𝑆) → Fun (inr ↾ dom 𝑆))
1614, 15ax-mp 5 . . . . 5 Fun (inr ↾ dom 𝑆)
17 djuinj.s . . . . 5 (𝜑 → Fun 𝑆)
18 funco 5158 . . . . 5 ((Fun (inr ↾ dom 𝑆) ∧ Fun 𝑆) → Fun ((inr ↾ dom 𝑆) ∘ 𝑆))
1916, 17, 18sylancr 410 . . . 4 (𝜑 → Fun ((inr ↾ dom 𝑆) ∘ 𝑆))
20 cnvco 4719 . . . . 5 (𝑆(inr ↾ dom 𝑆)) = ((inr ↾ dom 𝑆) ∘ 𝑆)
2120funeqi 5139 . . . 4 (Fun (𝑆(inr ↾ dom 𝑆)) ↔ Fun ((inr ↾ dom 𝑆) ∘ 𝑆))
2219, 21sylibr 133 . . 3 (𝜑 → Fun (𝑆(inr ↾ dom 𝑆)))
23 df-rn 4545 . . . . . . 7 ran (𝑅(inl ↾ dom 𝑅)) = dom (𝑅(inl ↾ dom 𝑅))
24 rncoss 4804 . . . . . . 7 ran (𝑅(inl ↾ dom 𝑅)) ⊆ ran 𝑅
2523, 24eqsstrri 3125 . . . . . 6 dom (𝑅(inl ↾ dom 𝑅)) ⊆ ran 𝑅
26 df-rn 4545 . . . . . . 7 ran (𝑆(inr ↾ dom 𝑆)) = dom (𝑆(inr ↾ dom 𝑆))
27 rncoss 4804 . . . . . . 7 ran (𝑆(inr ↾ dom 𝑆)) ⊆ ran 𝑆
2826, 27eqsstrri 3125 . . . . . 6 dom (𝑆(inr ↾ dom 𝑆)) ⊆ ran 𝑆
29 ss2in 3299 . . . . . 6 ((dom (𝑅(inl ↾ dom 𝑅)) ⊆ ran 𝑅 ∧ dom (𝑆(inr ↾ dom 𝑆)) ⊆ ran 𝑆) → (dom (𝑅(inl ↾ dom 𝑅)) ∩ dom (𝑆(inr ↾ dom 𝑆))) ⊆ (ran 𝑅 ∩ ran 𝑆))
3025, 28, 29mp2an 422 . . . . 5 (dom (𝑅(inl ↾ dom 𝑅)) ∩ dom (𝑆(inr ↾ dom 𝑆))) ⊆ (ran 𝑅 ∩ ran 𝑆)
31 djuinj.disj . . . . 5 (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)
3230, 31sseqtrid 3142 . . . 4 (𝜑 → (dom (𝑅(inl ↾ dom 𝑅)) ∩ dom (𝑆(inr ↾ dom 𝑆))) ⊆ ∅)
33 ss0 3398 . . . 4 ((dom (𝑅(inl ↾ dom 𝑅)) ∩ dom (𝑆(inr ↾ dom 𝑆))) ⊆ ∅ → (dom (𝑅(inl ↾ dom 𝑅)) ∩ dom (𝑆(inr ↾ dom 𝑆))) = ∅)
3432, 33syl 14 . . 3 (𝜑 → (dom (𝑅(inl ↾ dom 𝑅)) ∩ dom (𝑆(inr ↾ dom 𝑆))) = ∅)
35 funun 5162 . . 3 (((Fun (𝑅(inl ↾ dom 𝑅)) ∧ Fun (𝑆(inr ↾ dom 𝑆))) ∧ (dom (𝑅(inl ↾ dom 𝑅)) ∩ dom (𝑆(inr ↾ dom 𝑆))) = ∅) → Fun ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆))))
3611, 22, 34, 35syl21anc 1215 . 2 (𝜑 → Fun ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆))))
37 df-djud 6981 . . . . 5 (𝑅d 𝑆) = ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆)))
3837cnveqi 4709 . . . 4 (𝑅d 𝑆) = ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆)))
39 cnvun 4939 . . . 4 ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆))) = ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆)))
4038, 39eqtri 2158 . . 3 (𝑅d 𝑆) = ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆)))
4140funeqi 5139 . 2 (Fun (𝑅d 𝑆) ↔ Fun ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆))))
4236, 41sylibr 133 1 (𝜑 → Fun (𝑅d 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  cun 3064  cin 3065  wss 3066  c0 3358  ccnv 4533  dom cdm 4534  ran crn 4535  cres 4536  ccom 4538  Fun wfun 5112  1-1wf1 5115  cdju 6915  inlcinl 6923  inrcinr 6924  d cdjud 6980
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 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-dju 6916  df-inl 6925  df-inr 6926  df-djud 6981
This theorem is referenced by: (None)
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