Theorem djulclb 7359

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 7355	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
2		1n0 6678	. . . . . . . . . 10 |- 1o =/= (/)
3	2	necomi 2499	. . . . . . . . 9 |- (/) =/= 1o
4		0ex 4242	. . . . . . . . . 10 |- (/) e. _V
5	4	elsn 3710	. . . . . . . . 9 |- ( (/) e. { 1o } <-> (/) = 1o )
6	3, 5	nemtbir 2503	. . . . . . . 8 |- -. (/) e. { 1o }
7	6	intnanr 938	. . . . . . 7 |- -. ( (/) e. { 1o } /\ C e. B )
8		opelxp 4784	. . . . . . 7 |- ( <. (/) , C >. e. ( { 1o } X. B ) <-> ( (/) e. { 1o } /\ C e. B ) )
9	7, 8	mtbir 678	. . . . . 6 |- -. <. (/) , C >. e. ( { 1o } X. B )
10		elex 2827	. . . . . . . . . . . 12 |- ( C e. V -> C e. _V )
11		opexg 4349	. . . . . . . . . . . . 13 |- ( ( (/) e. _V /\ C e. V ) -> <. (/) , C >. e. _V )
12	4, 11	mpan 424	. . . . . . . . . . . 12 |- ( C e. V -> <. (/) , C >. e. _V )
13		opeq2 3889	. . . . . . . . . . . . 13 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
14		df-inl 7351	. . . . . . . . . . . . 13 |- inl = ( x e. _V |-> <. (/) , x >. )
15	13, 14	fvmptg 5758	. . . . . . . . . . . 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 7342	. . . . . . . . . . . . 13 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
19	18	eleq2i 2301	. . . . . . . . . . . 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 2312	. . . . . . . . 9 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
23		elun 3364	. . . . . . . . 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 737	. . . . . . 7 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( <. (/) , C >. e. ( { 1o } X. B ) \/ <. (/) , C >. e. ( { (/) } X. A ) ) )
26	25	ord 732	. . . . . 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 4784	. . . . 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 716   = wceq 1398   e. wcel 2205   _Vcvv 2815   u. cun 3212   (/)c0 3512   {csn 3694   <.cop 3697   X. cxp 4752   ` cfv 5357   1oc1o 6653   ⊔ cdju 7341  inlcinl 7349

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-1o 6660  df-dju 7342  df-inl 7351

This theorem is referenced by:  exmidfodomrlemr  7518