Theorem eldju 7045

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 7043	. . . 4 |- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
2	1	eqcomi 2174	. . 3 |- ( A ⊔ B ) = ( ran (inl |` A ) u. ran (inr |` B ) )
3	2	eleq2i 2237	. 2 |- ( C e. ( A ⊔ B ) <-> C e. ( ran (inl |` A ) u. ran (inr |` B ) ) )
4		elun 3268	. . 3 |- ( C e. ( ran (inl |` A ) u. ran (inr |` B ) ) <-> ( C e. ran (inl |` A ) \/ C e. ran (inr |` B ) ) )
5		djulf1or 7033	. . . . . 6 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
6		f1ofn 5443	. . . . . 6 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) -> (inl |` A ) Fn A )
7		fvelrnb 5544	. . . . . 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 2172	. . . . . 6 |- ( ( (inl |` A ) ` x ) = C <-> C = ( (inl |` A ) ` x ) )
10	9	rexbii 2477	. . . . 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 7034	. . . . . 6 |- (inr |` B ) : B -1-1-onto-> ( { 1o } X. B )
13		f1ofn 5443	. . . . . 6 |- ( (inr |` B ) : B -1-1-onto-> ( { 1o } X. B ) -> (inr |` B ) Fn B )
14		fvelrnb 5544	. . . . . 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 2172	. . . . . 6 |- ( ( (inr |` B ) ` x ) = C <-> C = ( (inr |` B ) ` x ) )
17	16	rexbii 2477	. . . . 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 759	. . 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 703   = wceq 1348   e. wcel 2141   E.wrex 2449   u. cun 3119   (/)c0 3414   {csn 3583   X. cxp 4609   ran crn 4612   |` cres 4613   Fn wfn 5193   -1-1-onto->wf1o 5197   ` cfv 5198   1oc1o 6388   ⊔ cdju 7014  inlcinl 7022  inrcinr 7023

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025

This theorem is referenced by:  djur  7046  exmidfodomrlemreseldju  7177