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Theorem djuunr 7010
Description: The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
Assertion
Ref Expression
djuunr (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)

Proof of Theorem djuunr
StepHypRef Expression
1 djulf1or 7000 . . . 4 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
2 f1ofo 5421 . . . 4 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴):𝐴onto→({∅} × 𝐴))
3 forn 5395 . . . 4 ((inl ↾ 𝐴):𝐴onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴))
41, 2, 3mp2b 8 . . 3 ran (inl ↾ 𝐴) = ({∅} × 𝐴)
5 djurf1or 7001 . . . 4 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
6 f1ofo 5421 . . . 4 ((inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵):𝐵onto→({1o} × 𝐵))
7 forn 5395 . . . 4 ((inr ↾ 𝐵):𝐵onto→({1o} × 𝐵) → ran (inr ↾ 𝐵) = ({1o} × 𝐵))
85, 6, 7mp2b 8 . . 3 ran (inr ↾ 𝐵) = ({1o} × 𝐵)
94, 8uneq12i 3259 . 2 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
10 df-dju 6982 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
119, 10eqtr4i 2181 1 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1335  cun 3100  c0 3394  {csn 3560   × cxp 4584  ran crn 4587  cres 4588  ontowfo 5168  1-1-ontowf1o 5169  1oc1o 6356  cdju 6981  inlcinl 6989  inrcinr 6990
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-2nd 6089  df-1o 6363  df-dju 6982  df-inl 6991  df-inr 6992
This theorem is referenced by:  djuun  7011  eldju  7012  casedm  7030  djudm  7049
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