Theorem eldju 7033

Description: Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)

Assertion

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

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

Proof of Theorem eldju

Step	Hyp	Ref	Expression
1		djuunr 7031	. . . 4 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
2	1	eqcomi 2169	. . 3 |- ( A ⊔ B ) = ( ran (inl |` A ) u. ran (inr |` B ) )
3	2	eleq2i 2233	. 2 |- ( C e. ( A ⊔ B ) <-> C e. ( ran (inl |` A ) u. ran (inr |` B ) ) )
4		elun 3263	. . 3 |- ( C e. ( ran (inl |` A ) u. ran (inr |` B ) ) <-> ( C e. ran (inl |` A ) \/ C e. ran (inr |` B ) ) )
5		djulf1or 7021	. . . . . 6 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
6		f1ofn 5433	. . . . . 6 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) -> (inl |` A ) Fn A )
7		fvelrnb 5534	. . . . . 6 |- ( (inl |` A ) Fn A -> ( C e. ran (inl |` A ) <-> E. x e. A ( (inl |` A ) ` x ) = C ) )
8	5, 6, 7	mp2b 8	. . . . 5 |- ( C e. ran (inl |` A ) <-> E. x e. A ( (inl |` A ) ` x ) = C )
9		eqcom 2167	. . . . . 6 |- ( ( (inl |` A ) ` x ) = C <-> C = ( (inl |` A ) ` x ) )
10	9	rexbii 2473	. . . . 5 |- ( E. x e. A ( (inl |` A ) ` x ) = C <-> E. x e. A C = ( (inl |` A ) ` x ) )
11	8, 10	bitri 183	. . . 4 |- ( C e. ran (inl |` A ) <-> E. x e. A C = ( (inl |` A ) ` x ) )
12		djurf1or 7022	. . . . . 6 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
13		f1ofn 5433	. . . . . 6 |- ( (inr |` B ) : B -1-1-onto-> ( { 1o } X. B ) -> (inr |` B ) Fn B )
14		fvelrnb 5534	. . . . . 6 |- ( (inr |` B ) Fn B -> ( C e. ran (inr |` B ) <-> E. x e. B ( (inr |` B ) ` x ) = C ) )
15	12, 13, 14	mp2b 8	. . . . 5 |- ( C e. ran (inr |` B ) <-> E. x e. B ( (inr |` B ) ` x ) = C )
16		eqcom 2167	. . . . . 6 |- ( ( (inr |` B ) ` x ) = C <-> C = ( (inr |` B ) ` x ) )
17	16	rexbii 2473	. . . . 5 |- ( E. x e. B ( (inr |` B ) ` x ) = C <-> E. x e. B C = ( (inr |` B ) ` x ) )
18	15, 17	bitri 183	. . . 4 |- ( C e. ran (inr |` B ) <-> E. x e. B C = ( (inr |` B ) ` x ) )
19	11, 18	orbi12i 754	. . 3 |- ( ( C e. ran (inl |` A ) \/ C e. ran (inr |` B ) ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )
20	4, 19	bitri 183	. 2 |- ( C e. ( ran (inl |` A ) u. ran (inr |` B ) ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )
21	3, 20	bitri 183	1 |- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )

Syntax hints:   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   E.wrex 2445   u. cun 3114   (/)c0 3409   {csn 3576   X. cxp 4602   ran crn 4605   |` cres 4606   Fn wfn 5183   -1-1-onto->wf1o 5187   ` cfv 5188   1oc1o 6377   ⊔ cdju 7002  inlcinl 7010  inrcinr 7011

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013

This theorem is referenced by:  djur  7034  exmidfodomrlemreseldju  7156