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Theorem djuinj 6765
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 6732 . . . . . . 7 (inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅𝐴)
2 f1fun 5203 . . . . . . 7 ((inl ↾ dom 𝑅):dom 𝑅1-1→(dom 𝑅𝐴) → Fun (inl ↾ dom 𝑅))
31, 2ax-mp 7 . . . . . 6 Fun (inl ↾ dom 𝑅)
4 funcnvcnv 5059 . . . . . 6 (Fun (inl ↾ dom 𝑅) → Fun (inl ↾ dom 𝑅))
53, 4ax-mp 7 . . . . 5 Fun (inl ↾ dom 𝑅)
6 djuinj.r . . . . 5 (𝜑 → Fun 𝑅)
7 funco 5040 . . . . 5 ((Fun (inl ↾ dom 𝑅) ∧ Fun 𝑅) → Fun ((inl ↾ dom 𝑅) ∘ 𝑅))
85, 6, 7sylancr 405 . . . 4 (𝜑 → Fun ((inl ↾ dom 𝑅) ∘ 𝑅))
9 cnvco 4609 . . . . 5 (𝑅(inl ↾ dom 𝑅)) = ((inl ↾ dom 𝑅) ∘ 𝑅)
109funeqi 5022 . . . 4 (Fun (𝑅(inl ↾ dom 𝑅)) ↔ Fun ((inl ↾ dom 𝑅) ∘ 𝑅))
118, 10sylibr 132 . . 3 (𝜑 → Fun (𝑅(inl ↾ dom 𝑅)))
12 inrresf1 6733 . . . . . . 7 (inr ↾ dom 𝑆):dom 𝑆1-1→(𝐴 ⊔ dom 𝑆)
13 f1fun 5203 . . . . . . 7 ((inr ↾ dom 𝑆):dom 𝑆1-1→(𝐴 ⊔ dom 𝑆) → Fun (inr ↾ dom 𝑆))
1412, 13ax-mp 7 . . . . . 6 Fun (inr ↾ dom 𝑆)
15 funcnvcnv 5059 . . . . . 6 (Fun (inr ↾ dom 𝑆) → Fun (inr ↾ dom 𝑆))
1614, 15ax-mp 7 . . . . 5 Fun (inr ↾ dom 𝑆)
17 djuinj.s . . . . 5 (𝜑 → Fun 𝑆)
18 funco 5040 . . . . 5 ((Fun (inr ↾ dom 𝑆) ∧ Fun 𝑆) → Fun ((inr ↾ dom 𝑆) ∘ 𝑆))
1916, 17, 18sylancr 405 . . . 4 (𝜑 → Fun ((inr ↾ dom 𝑆) ∘ 𝑆))
20 cnvco 4609 . . . . 5 (𝑆(inr ↾ dom 𝑆)) = ((inr ↾ dom 𝑆) ∘ 𝑆)
2120funeqi 5022 . . . 4 (Fun (𝑆(inr ↾ dom 𝑆)) ↔ Fun ((inr ↾ dom 𝑆) ∘ 𝑆))
2219, 21sylibr 132 . . 3 (𝜑 → Fun (𝑆(inr ↾ dom 𝑆)))
23 df-rn 4439 . . . . . . 7 ran (𝑅(inl ↾ dom 𝑅)) = dom (𝑅(inl ↾ dom 𝑅))
24 rncoss 4691 . . . . . . 7 ran (𝑅(inl ↾ dom 𝑅)) ⊆ ran 𝑅
2523, 24eqsstr3i 3055 . . . . . 6 dom (𝑅(inl ↾ dom 𝑅)) ⊆ ran 𝑅
26 df-rn 4439 . . . . . . 7 ran (𝑆(inr ↾ dom 𝑆)) = dom (𝑆(inr ↾ dom 𝑆))
27 rncoss 4691 . . . . . . 7 ran (𝑆(inr ↾ dom 𝑆)) ⊆ ran 𝑆
2826, 27eqsstr3i 3055 . . . . . 6 dom (𝑆(inr ↾ dom 𝑆)) ⊆ ran 𝑆
29 ss2in 3225 . . . . . 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 417 . . . . 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, 31syl5sseq 3072 . . . 4 (𝜑 → (dom (𝑅(inl ↾ dom 𝑅)) ∩ dom (𝑆(inr ↾ dom 𝑆))) ⊆ ∅)
33 ss0 3320 . . . 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 5044 . . 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 1173 . 2 (𝜑 → Fun ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆))))
37 df-djud 6762 . . . . 5 (𝑅d 𝑆) = ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆)))
3837cnveqi 4599 . . . 4 (𝑅d 𝑆) = ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆)))
39 cnvun 4824 . . . 4 ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆))) = ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆)))
4038, 39eqtri 2108 . . 3 (𝑅d 𝑆) = ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆)))
4140funeqi 5022 . 2 (Fun (𝑅d 𝑆) ↔ Fun ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆))))
4236, 41sylibr 132 1 (𝜑 → Fun (𝑅d 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cun 2995  cin 2996  wss 2997  c0 3284  ccnv 4427  dom cdm 4428  ran crn 4429  cres 4430  ccom 4432  Fun wfun 4996  1-1wf1 4999  cdju 6709  inlcinl 6716  inrcinr 6717  d cdjud 6761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-1st 5893  df-2nd 5894  df-1o 6163  df-dju 6710  df-inl 6718  df-inr 6719  df-djud 6762
This theorem is referenced by: (None)
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