Theorem djuun 7169

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, 9-Jul-2023.)

Assertion

Ref	Expression
djuun	|- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )

Proof of Theorem djuun

Step	Hyp	Ref	Expression
1		df-ima 4688	. . 3 |- (inl " A ) = ran (inl |` A )
2		df-ima 4688	. . 3 |- (inr " B ) = ran (inr |` B )
3	1, 2	uneq12i 3325	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( ran (inl |` A ) u. ran (inr |` B ) )
4		djuunr 7168	. 2 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
5	3, 4	eqtri 2226	1 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )

Syntax hints:   = wceq 1373   u. cun 3164   ran crn 4676   |` cres 4677   "cima 4678   ⊔ cdju 7139  inlcinl 7147  inrcinr 7148

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6226  df-2nd 6227  df-1o 6502  df-dju 7140  df-inl 7149  df-inr 7150

This theorem is referenced by:  endjusym  7198  ctssdccl  7213  dju1p1e2  7305  endjudisj  7322  djuen  7323