Theorem djulclb 7159

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 7155	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
2		1n0 6520	. . . . . . . . . 10 |- 1o =/= (/)
3	2	necomi 2461	. . . . . . . . 9 |- (/) =/= 1o
4		0ex 4172	. . . . . . . . . 10 |- (/) e. _V
5	4	elsn 3649	. . . . . . . . 9 |- ( (/) e. { 1o } <-> (/) = 1o )
6	3, 5	nemtbir 2465	. . . . . . . 8 |- -. (/) e. { 1o }
7	6	intnanr 932	. . . . . . 7 |- -. ( (/) e. { 1o } /\ C e. B )
8		opelxp 4706	. . . . . . 7 |- ( <. (/) , C >. e. ( { 1o } X. B ) <-> ( (/) e. { 1o } /\ C e. B ) )
9	7, 8	mtbir 673	. . . . . 6 |- -. <. (/) , C >. e. ( { 1o } X. B )
10		elex 2783	. . . . . . . . . . . 12 |- ( C e. V -> C e. _V )
11		opexg 4273	. . . . . . . . . . . . 13 |- ( ( (/) e. _V /\ C e. V ) -> <. (/) , C >. e. _V )
12	4, 11	mpan 424	. . . . . . . . . . . 12 |- ( C e. V -> <. (/) , C >. e. _V )
13		opeq2 3820	. . . . . . . . . . . . 13 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
14		df-inl 7151	. . . . . . . . . . . . 13 |- inl = ( x e. _V |-> <. (/) , x >. )
15	13, 14	fvmptg 5657	. . . . . . . . . . . 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 7142	. . . . . . . . . . . . 13 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
19	18	eleq2i 2272	. . . . . . . . . . . 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 2283	. . . . . . . . 9 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
23		elun 3314	. . . . . . . . 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 731	. . . . . . 7 |- ( ( C e. V /\ (inl ` C ) e. ( A ⊔ B ) ) -> ( <. (/) , C >. e. ( { 1o } X. B ) \/ <. (/) , C >. e. ( { (/) } X. A ) ) )
26	25	ord 726	. . . . . 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 4706	. . . . 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 710   = wceq 1373   e. wcel 2176   _Vcvv 2772   u. cun 3164   (/)c0 3460   {csn 3633   <.cop 3636   X. cxp 4674   ` cfv 5272   1oc1o 6497   ⊔ cdju 7141  inlcinl 7149

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-suc 4419  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-1o 6504  df-dju 7142  df-inl 7151

This theorem is referenced by:  exmidfodomrlemr  7312