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Theorem djuunr 6917
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 )  u. 
ran  (inr  |`  B ) )  =  ( A B )

Proof of Theorem djuunr
StepHypRef Expression
1 djulf1or 6907 . . . 4  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
2 f1ofo 5340 . . . 4  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  ->  (inl  |`  A ) : A -onto-> ( {
(/) }  X.  A
) )
3 forn 5316 . . . 4  |-  ( (inl  |`  A ) : A -onto->
( { (/) }  X.  A )  ->  ran  (inl  |`  A )  =  ( { (/) }  X.  A ) )
41, 2, 3mp2b 8 . . 3  |-  ran  (inl  |`  A )  =  ( { (/) }  X.  A
)
5 djurf1or 6908 . . . 4  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
6 f1ofo 5340 . . . 4  |-  ( (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B
)  ->  (inr  |`  B ) : B -onto-> ( { 1o }  X.  B
) )
7 forn 5316 . . . 4  |-  ( (inr  |`  B ) : B -onto->
( { 1o }  X.  B )  ->  ran  (inr  |`  B )  =  ( { 1o }  X.  B ) )
85, 6, 7mp2b 8 . . 3  |-  ran  (inr  |`  B )  =  ( { 1o }  X.  B )
94, 8uneq12i 3196 . 2  |-  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) )  =  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  B
) )
10 df-dju 6889 . 2  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
119, 10eqtr4i 2139 1  |-  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) )  =  ( A B )
Colors of variables: wff set class
Syntax hints:    = wceq 1314    u. cun 3037   (/)c0 3331   {csn 3495    X. cxp 4505   ran crn 4508    |` cres 4509   -onto->wfo 5089   -1-1-onto->wf1o 5090   1oc1o 6272   ⊔ cdju 6888  inlcinl 6896  inrcinr 6897
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899
This theorem is referenced by:  djuun  6918  eldju  6919  casedm  6937  djudm  6956
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