Theorem djulcl 7218

Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)

Assertion

Ref	Expression
djulcl	|- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )

Proof of Theorem djulcl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2811	. . 3 |- ( C e. A -> C e. _V )
2		0ex 4211	. . . . 5 |- (/) e. _V
3	2	snid 3697	. . . 4 |- (/) e. { (/) }
4		opelxpi 4751	. . . 4 |- ( ( (/) e. { (/) } /\ C e. A ) -> <. (/) , C >. e. ( { (/) } X. A ) )
5	3, 4	mpan 424	. . 3 |- ( C e. A -> <. (/) , C >. e. ( { (/) } X. A ) )
6		opeq2 3858	. . . 4 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
7		df-inl 7214	. . . 4 |- inl = ( x e. _V |-> <. (/) , x >. )
8	6, 7	fvmptg 5710	. . 3 |- ( ( C e. _V /\ <. (/) , C >. e. ( { (/) } X. A ) ) -> (inl ` C ) = <. (/) , C >. )
9	1, 5, 8	syl2anc 411	. 2 |- ( C e. A -> (inl ` C ) = <. (/) , C >. )
10		elun1 3371	. . . 4 |- ( <. (/) , C >. e. ( { (/) } X. A ) -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
11	5, 10	syl 14	. . 3 |- ( C e. A -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
12		df-dju 7205	. . 3 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
13	11, 12	eleqtrrdi 2323	. 2 |- ( C e. A -> <. (/) , C >. e. ( A ⊔ B ) )
14	9, 13	eqeltrd 2306	1 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   u. cun 3195   (/)c0 3491   {csn 3666   <.cop 3669   X. cxp 4717   ` cfv 5318   1oc1o 6555   ⊔ cdju 7204  inlcinl 7212

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-dju 7205  df-inl 7214

This theorem is referenced by:  djulclb  7222  updjudhcoinlf  7247  omp1eomlem  7261  difinfsnlem  7266  difinfsn  7267  ctmlemr  7275  ctm  7276  ctssdclemn0  7277  ctssdccl  7278  fodju0  7314  exmidfodomrlemr  7380  exmidfodomrlemrALT  7381  subctctexmid  16366