Theorem eldju 7196

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 7194	. . . 4 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
2	1	eqcomi 2211	. . 3 |- ( A ⊔ B ) = ( ran (inl |` A ) u. ran (inr |` B ) )
3	2	eleq2i 2274	. 2 |- ( C e. ( A ⊔ B ) <-> C e. ( ran (inl |` A ) u. ran (inr |` B ) ) )
4		elun 3322	. . 3 |- ( C e. ( ran (inl |` A ) u. ran (inr |` B ) ) <-> ( C e. ran (inl |` A ) \/ C e. ran (inr |` B ) ) )
5		djulf1or 7184	. . . . . 6 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
6		f1ofn 5545	. . . . . 6 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) -> (inl |` A ) Fn A )
7		fvelrnb 5649	. . . . . 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 2209	. . . . . 6 |- ( ( (inl |` A ) ` x ) = C <-> C = ( (inl |` A ) ` x ) )
10	9	rexbii 2515	. . . . 5 |- ( E. x e. A ( (inl |` A ) ` x ) = C <-> E. x e. A C = ( (inl |` A ) ` x ) )
11	8, 10	bitri 184	. . . 4 |- ( C e. ran (inl |` A ) <-> E. x e. A C = ( (inl |` A ) ` x ) )
12		djurf1or 7185	. . . . . 6 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
13		f1ofn 5545	. . . . . 6 |- ( (inr |` B ) : B -1-1-onto-> ( { 1o } X. B ) -> (inr |` B ) Fn B )
14		fvelrnb 5649	. . . . . 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 2209	. . . . . 6 |- ( ( (inr |` B ) ` x ) = C <-> C = ( (inr |` B ) ` x ) )
17	16	rexbii 2515	. . . . 5 |- ( E. x e. B ( (inr |` B ) ` x ) = C <-> E. x e. B C = ( (inr |` B ) ` x ) )
18	15, 17	bitri 184	. . . 4 |- ( C e. ran (inr |` B ) <-> E. x e. B C = ( (inr |` B ) ` x ) )
19	11, 18	orbi12i 766	. . 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 184	. 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 184	1 |- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )

Syntax hints:   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2178   E.wrex 2487   u. cun 3172   (/)c0 3468   {csn 3643   X. cxp 4691   ran crn 4694   |` cres 4695   Fn wfn 5285   -1-1-onto->wf1o 5289   ` cfv 5290   1oc1o 6518   ⊔ cdju 7165  inlcinl 7173  inrcinr 7174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250  df-1o 6525  df-dju 7166  df-inl 7175  df-inr 7176

This theorem is referenced by:  djur  7197  exmidfodomrlemreseldju  7339