Theorem djur 7267

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 7266	. 2 |- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )
2		fvres 5663	. . . . 5 |- ( x e. A -> ( (inl |` A ) ` x ) = (inl ` x ) )
3	2	eqeq2d 2243	. . . 4 |- ( x e. A -> ( C = ( (inl |` A ) ` x ) <-> C = (inl ` x ) ) )
4	3	rexbiia 2547	. . 3 |- ( E. x e. A C = ( (inl |` A ) ` x ) <-> E. x e. A C = (inl ` x ) )
5		fvres 5663	. . . . 5 |- ( x e. B -> ( (inr |` B ) ` x ) = (inr ` x ) )
6	5	eqeq2d 2243	. . . 4 |- ( x e. B -> ( C = ( (inr |` B ) ` x ) <-> C = (inr ` x ) ) )
7	6	rexbiia 2547	. . 3 |- ( E. x e. B C = ( (inr |` B ) ` x ) <-> E. x e. B C = (inr ` x ) )
8	4, 7	orbi12i 771	. 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 715   = wceq 1397   e. wcel 2202   E.wrex 2511   |` cres 4727   ` cfv 5326   ⊔ cdju 7235  inlcinl 7243  inrcinr 7244

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  djuss  7268  updjud  7280  omp1eomlem  7292  0ct  7305  ctmlemr  7306  ctssdclemn0  7308  fodjuomnilemdc  7342  exmidfodomrlemeldju  7409