Theorem djur 6962

Description: A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)

Assertion

Ref	Expression
djur	|- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = (inl ` x ) \/ E. x e. B C = (inr ` x ) ) )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem djur

Step	Hyp	Ref	Expression
1		eldju 6961	. 2 |- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )
2		fvres 5453	. . . . 5 |- ( x e. A -> ( (inl |` A ) ` x ) = (inl ` x ) )
3	2	eqeq2d 2152	. . . 4 |- ( x e. A -> ( C = ( (inl |` A ) ` x ) <-> C = (inl ` x ) ) )
4	3	rexbiia 2453	. . 3 |- ( E. x e. A C = ( (inl |` A ) ` x ) <-> E. x e. A C = (inl ` x ) )
5		fvres 5453	. . . . 5 |- ( x e. B -> ( (inr |` B ) ` x ) = (inr ` x ) )
6	5	eqeq2d 2152	. . . 4 |- ( x e. B -> ( C = ( (inr |` B ) ` x ) <-> C = (inr ` x ) ) )
7	6	rexbiia 2453	. . 3 |- ( E. x e. B C = ( (inr |` B ) ` x ) <-> E. x e. B C = (inr ` x ) )
8	4, 7	orbi12i 754	. 2 |- ( ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) <-> ( E. x e. A C = (inl ` x ) \/ E. x e. B C = (inr ` x ) ) )
9	1, 8	bitri 183	1 |- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = (inl ` x ) \/ E. x e. B C = (inr ` x ) ) )

Syntax hints:   <-> wb 104   \/ wo 698   = wceq 1332   e. wcel 1481   E.wrex 2418   |` cres 4549   ` cfv 5131   ⊔ cdju 6930  inlcinl 6938  inrcinr 6939

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-dju 6931  df-inl 6940  df-inr 6941

This theorem is referenced by:  djuss  6963  updjud  6975  omp1eomlem  6987  0ct  7000  ctmlemr  7001  ctssdclemn0  7003  fodjuomnilemdc  7024  exmidfodomrlemeldju  7072