Theorem djulclb 6892

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 6888	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
2		1n0 6283	. . . . . . . . . 10 |- 1o =/= (/)
3	2	necomi 2367	. . . . . . . . 9 |- (/) =/= 1o
4		0ex 4015	. . . . . . . . . 10 |- (/) e. _V
5	4	elsn 3509	. . . . . . . . 9 |- ( (/) e. { 1o } <-> (/) = 1o )
6	3, 5	nemtbir 2371	. . . . . . . 8 |- -. (/) e. { 1o }
7	6	intnanr 898	. . . . . . 7 |- -. ( (/) e. { 1o } /\ C e. B )
8		opelxp 4529	. . . . . . 7 |- ( <. (/) , C >. e. ( { 1o } X. B ) <-> ( (/) e. { 1o } /\ C e. B ) )
9	7, 8	mtbir 643	. . . . . 6 |- -. <. (/) , C >. e. ( { 1o } X. B )
10		elex 2668	. . . . . . . . . . . 12 |- ( C e. V -> C e. _V )
11		opexg 4110	. . . . . . . . . . . . 13 |- ( ( (/) e. _V /\ C e. V ) -> <. (/) , C >. e. _V )
12	4, 11	mpan 418	. . . . . . . . . . . 12 |- ( C e. V -> <. (/) , C >. e. _V )
13		opeq2 3672	. . . . . . . . . . . . 13 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
14		df-inl 6884	. . . . . . . . . . . . 13 |- inl = ( x e. _V |-> <. (/) , x >. )
15	13, 14	fvmptg 5451	. . . . . . . . . . . 12 |- ( ( C e. _V /\ <. (/) , C >. e. _V ) -> (inl ` C ) = <. (/) , C >. )
16	10, 12, 15	syl2anc 406	. . . . . . . . . . 11 |- ( C e. V -> (inl ` C ) = <. (/) , C >. )
17	16	adantr 272	. . . . . . . . . 10 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> (inl ` C ) = <. (/) , C >. )
18		df-dju 6875	. . . . . . . . . . . . 13 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
19	18	eleq2i 2181	. . . . . . . . . . . 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 273	. . . . . . . . . 10 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> (inl ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
22	17, 21	eqeltrrd 2192	. . . . . . . . 9 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
23		elun 3183	. . . . . . . . 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 701	. . . . . . 7 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( <. (/) , C >. e. ( { 1o } X. B ) \/ <. (/) , C >. e. ( { (/) } X. A ) ) )
26	25	ord 696	. . . . . 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 4529	. . . . 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 680   = wceq 1314   e. wcel 1463   _Vcvv 2657   u. cun 3035   (/)c0 3329   {csn 3493   <.cop 3496   X. cxp 4497   ` cfv 5081   1oc1o 6260   ⊔ cdju 6874  inlcinl 6882

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-suc 4253  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-1o 6267  df-dju 6875  df-inl 6884

This theorem is referenced by:  exmidfodomrlemr  7006