Theorem djulclb 7020

Description: Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)

Assertion

Ref	Expression
djulclb	|- ( C e. V -> ( C e. A <-> (inl ` C ) e. ( A ⊔ B ) ) )

Proof of Theorem djulclb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djulcl 7016	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
2		1n0 6400	. . . . . . . . . 10 |- 1o =/= (/)
3	2	necomi 2421	. . . . . . . . 9 |- (/) =/= 1o
4		0ex 4109	. . . . . . . . . 10 |- (/) e. _V
5	4	elsn 3592	. . . . . . . . 9 |- ( (/) e. { 1o } <-> (/) = 1o )
6	3, 5	nemtbir 2425	. . . . . . . 8 |- -. (/) e. { 1o }
7	6	intnanr 920	. . . . . . 7 |- -. ( (/) e. { 1o } /\ C e. B )
8		opelxp 4634	. . . . . . 7 |- ( <. (/) , C >. e. ( { 1o } X. B ) <-> ( (/) e. { 1o } /\ C e. B ) )
9	7, 8	mtbir 661	. . . . . 6 |- -. <. (/) , C >. e. ( { 1o } X. B )
10		elex 2737	. . . . . . . . . . . 12 |- ( C e. V -> C e. _V )
11		opexg 4206	. . . . . . . . . . . . 13 |- ( ( (/) e. _V /\ C e. V ) -> <. (/) , C >. e. _V )
12	4, 11	mpan 421	. . . . . . . . . . . 12 |- ( C e. V -> <. (/) , C >. e. _V )
13		opeq2 3759	. . . . . . . . . . . . 13 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
14		df-inl 7012	. . . . . . . . . . . . 13 |- inl = ( x e. _V |-> <. (/) , x >. )
15	13, 14	fvmptg 5562	. . . . . . . . . . . 12 |- ( ( C e. _V /\ <. (/) , C >. e. _V ) -> (inl ` C ) = <. (/) , C >. )
16	10, 12, 15	syl2anc 409	. . . . . . . . . . 11 |- ( C e. V -> (inl ` C ) = <. (/) , C >. )
17	16	adantr 274	. . . . . . . . . 10 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> (inl ` C ) = <. (/) , C >. )
18		df-dju 7003	. . . . . . . . . . . . 13 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
19	18	eleq2i 2233	. . . . . . . . . . . 12 |- ( (inl ` C ) e. ( A ⊔ B ) <-> (inl ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
20	19	biimpi 119	. . . . . . . . . . 11 |- ( (inl ` C ) e. ( A ⊔ B ) -> (inl ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
21	20	adantl 275	. . . . . . . . . 10 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> (inl ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
22	17, 21	eqeltrrd 2244	. . . . . . . . 9 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
23		elun 3263	. . . . . . . . 9 |- ( <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) <-> ( <. (/) , C >. e. ( { (/) } X. A ) \/ <. (/) , C >. e. ( { 1o } X. B ) ) )
24	22, 23	sylib 121	. . . . . . . 8 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( <. (/) , C >. e. ( { (/) } X. A ) \/ <. (/) , C >. e. ( { 1o } X. B ) ) )
25	24	orcomd 719	. . . . . . 7 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( <. (/) , C >. e. ( { 1o } X. B ) \/ <. (/) , C >. e. ( { (/) } X. A ) ) )
26	25	ord 714	. . . . . 6 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( -. <. (/) , C >. e. ( { 1o } X. B ) -> <. (/) , C >. e. ( { (/) } X. A ) ) )
27	9, 26	mpi 15	. . . . 5 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> <. (/) , C >. e. ( { (/) } X. A ) )
28		opelxp 4634	. . . . 5 |- ( <. (/) , C >. e. ( { (/) } X. A ) <-> ( (/) e. { (/) } /\ C e. A ) )
29	27, 28	sylib 121	. . . 4 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( (/) e. { (/) } /\ C e. A ) )
30	29	simprd 113	. . 3 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> C e. A )
31	30	ex 114	. 2 |- ( C e. V -> ( (inl ` C ) e. ( A ⊔ B ) -> C e. A ) )
32	1, 31	impbid2 142	1 |- ( C e. V -> ( C e. A <-> (inl ` C ) e. ( A ⊔ B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   _Vcvv 2726   u. cun 3114   (/)c0 3409   {csn 3576   <.cop 3579   X. cxp 4602   ` cfv 5188   1oc1o 6377   ⊔ cdju 7002  inlcinl 7010

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-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187

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-ne 2337  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-id 4271  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-1o 6384  df-dju 7003  df-inl 7012

This theorem is referenced by:  exmidfodomrlemr  7158