Theorem djur 7135

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 7134	. 2 |- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )
2		fvres 5582	. . . . 5 |- ( x e. A -> ( (inl |` A ) ` x ) = (inl ` x ) )
3	2	eqeq2d 2208	. . . 4 |- ( x e. A -> ( C = ( (inl |` A ) ` x ) <-> C = (inl ` x ) ) )
4	3	rexbiia 2512	. . 3 |- ( E. x e. A C = ( (inl |` A ) ` x ) <-> E. x e. A C = (inl ` x ) )
5		fvres 5582	. . . . 5 |- ( x e. B -> ( (inr |` B ) ` x ) = (inr ` x ) )
6	5	eqeq2d 2208	. . . 4 |- ( x e. B -> ( C = ( (inr |` B ) ` x ) <-> C = (inr ` x ) ) )
7	6	rexbiia 2512	. . 3 |- ( E. x e. B C = ( (inr |` B ) ` x ) <-> E. x e. B C = (inr ` x ) )
8	4, 7	orbi12i 765	. 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 709   = wceq 1364   e. wcel 2167   E.wrex 2476   |` cres 4665   ` cfv 5258   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114

This theorem is referenced by:  djuss  7136  updjud  7148  omp1eomlem  7160  0ct  7173  ctmlemr  7174  ctssdclemn0  7176  fodjuomnilemdc  7210  exmidfodomrlemeldju  7266