Theorem djur 7236

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 7235	. 2 |- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )
2		fvres 5651	. . . . 5 |- ( x e. A -> ( (inl |` A ) ` x ) = (inl ` x ) )
3	2	eqeq2d 2241	. . . 4 |- ( x e. A -> ( C = ( (inl |` A ) ` x ) <-> C = (inl ` x ) ) )
4	3	rexbiia 2545	. . 3 |- ( E. x e. A C = ( (inl |` A ) ` x ) <-> E. x e. A C = (inl ` x ) )
5		fvres 5651	. . . . 5 |- ( x e. B -> ( (inr |` B ) ` x ) = (inr ` x ) )
6	5	eqeq2d 2241	. . . 4 |- ( x e. B -> ( C = ( (inr |` B ) ` x ) <-> C = (inr ` x ) ) )
7	6	rexbiia 2545	. . 3 |- ( E. x e. B C = ( (inr |` B ) ` x ) <-> E. x e. B C = (inr ` x ) )
8	4, 7	orbi12i 769	. 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 713   = wceq 1395   e. wcel 2200   E.wrex 2509   |` cres 4721   ` cfv 5318   ⊔ cdju 7204  inlcinl 7212  inrcinr 7213

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6286  df-2nd 6287  df-1o 6562  df-dju 7205  df-inl 7214  df-inr 7215

This theorem is referenced by:  djuss  7237  updjud  7249  omp1eomlem  7261  0ct  7274  ctmlemr  7275  ctssdclemn0  7277  fodjuomnilemdc  7311  exmidfodomrlemeldju  7377