Theorem djulclb 7114

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 7110	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
2		1n0 6485	. . . . . . . . . 10 |- 1o =/= (/)
3	2	necomi 2449	. . . . . . . . 9 |- (/) =/= 1o
4		0ex 4156	. . . . . . . . . 10 |- (/) e. _V
5	4	elsn 3634	. . . . . . . . 9 |- ( (/) e. { 1o } <-> (/) = 1o )
6	3, 5	nemtbir 2453	. . . . . . . 8 |- -. (/) e. { 1o }
7	6	intnanr 931	. . . . . . 7 |- -. ( (/) e. { 1o } /\ C e. B )
8		opelxp 4689	. . . . . . 7 |- ( <. (/) , C >. e. ( { 1o } X. B ) <-> ( (/) e. { 1o } /\ C e. B ) )
9	7, 8	mtbir 672	. . . . . 6 |- -. <. (/) , C >. e. ( { 1o } X. B )
10		elex 2771	. . . . . . . . . . . 12 |- ( C e. V -> C e. _V )
11		opexg 4257	. . . . . . . . . . . . 13 |- ( ( (/) e. _V /\ C e. V ) -> <. (/) , C >. e. _V )
12	4, 11	mpan 424	. . . . . . . . . . . 12 |- ( C e. V -> <. (/) , C >. e. _V )
13		opeq2 3805	. . . . . . . . . . . . 13 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
14		df-inl 7106	. . . . . . . . . . . . 13 |- inl = ( x e. _V |-> <. (/) , x >. )
15	13, 14	fvmptg 5633	. . . . . . . . . . . 12 |- ( ( C e. _V /\ <. (/) , C >. e. _V ) -> (inl ` C ) = <. (/) , C >. )
16	10, 12, 15	syl2anc 411	. . . . . . . . . . 11 |- ( C e. V -> (inl ` C ) = <. (/) , C >. )
17	16	adantr 276	. . . . . . . . . 10 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> (inl ` C ) = <. (/) , C >. )
18		df-dju 7097	. . . . . . . . . . . . 13 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
19	18	eleq2i 2260	. . . . . . . . . . . 12 |- ( (inl ` C ) e. ( A ⊔ B ) <-> (inl ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
20	19	biimpi 120	. . . . . . . . . . 11 |- ( (inl ` C ) e. ( A ⊔ B ) -> (inl ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
21	20	adantl 277	. . . . . . . . . 10 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> (inl ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
22	17, 21	eqeltrrd 2271	. . . . . . . . 9 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
23		elun 3300	. . . . . . . . 9 |- ( <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) <-> ( <. (/) , C >. e. ( { (/) } X. A ) \/ <. (/) , C >. e. ( { 1o } X. B ) ) )
24	22, 23	sylib 122	. . . . . . . 8 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( <. (/) , C >. e. ( { (/) } X. A ) \/ <. (/) , C >. e. ( { 1o } X. B ) ) )
25	24	orcomd 730	. . . . . . 7 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( <. (/) , C >. e. ( { 1o } X. B ) \/ <. (/) , C >. e. ( { (/) } X. A ) ) )
26	25	ord 725	. . . . . 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 4689	. . . . 5 |- ( <. (/) , C >. e. ( { (/) } X. A ) <-> ( (/) e. { (/) } /\ C e. A ) )
29	27, 28	sylib 122	. . . 4 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( (/) e. { (/) } /\ C e. A ) )
30	29	simprd 114	. . 3 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> C e. A )
31	30	ex 115	. 2 |- ( C e. V -> ( (inl ` C ) e. ( A ⊔ B ) -> C e. A ) )
32	1, 31	impbid2 143	1 |- ( C e. V -> ( C e. A <-> (inl ` C ) e. ( A ⊔ B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   _Vcvv 2760   u. cun 3151   (/)c0 3446   {csn 3618   <.cop 3621   X. cxp 4657   ` cfv 5254   1oc1o 6462   ⊔ cdju 7096  inlcinl 7104

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-1o 6469  df-dju 7097  df-inl 7106

This theorem is referenced by:  exmidfodomrlemr  7262