Theorem djur 7058

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 7057	. 2 |- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )
2		fvres 5531	. . . . 5 |- ( x e. A -> ( (inl |` A ) ` x ) = (inl ` x ) )
3	2	eqeq2d 2187	. . . 4 |- ( x e. A -> ( C = ( (inl |` A ) ` x ) <-> C = (inl ` x ) ) )
4	3	rexbiia 2490	. . 3 |- ( E. x e. A C = ( (inl |` A ) ` x ) <-> E. x e. A C = (inl ` x ) )
5		fvres 5531	. . . . 5 |- ( x e. B -> ( (inr |` B ) ` x ) = (inr ` x ) )
6	5	eqeq2d 2187	. . . 4 |- ( x e. B -> ( C = ( (inr |` B ) ` x ) <-> C = (inr ` x ) ) )
7	6	rexbiia 2490	. . 3 |- ( E. x e. B C = ( (inr |` B ) ` x ) <-> E. x e. B C = (inr ` x ) )
8	4, 7	orbi12i 764	. 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 184	1 |- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = (inl ` x ) \/ E. x e. B C = (inr ` x ) ) )

Syntax hints:   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2146   E.wrex 2454   |` cres 4622   ` cfv 5208   ⊔ cdju 7026  inlcinl 7034  inrcinr 7035

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-dju 7027  df-inl 7036  df-inr 7037

This theorem is referenced by:  djuss  7059  updjud  7071  omp1eomlem  7083  0ct  7096  ctmlemr  7097  ctssdclemn0  7099  fodjuomnilemdc  7132  exmidfodomrlemeldju  7188