Theorem djulclr 7239

Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)

Assertion

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

Proof of Theorem djulclr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvres 5659	. 2 |- ( C e. A -> ( (inl |` A ) ` C ) = (inl ` C ) )
2		elex 2812	. . . 4 |- ( C e. A -> C e. _V )
3		0ex 4214	. . . . . 6 |- (/) e. _V
4	3	snid 3698	. . . . 5 |- (/) e. { (/) }
5		opelxpi 4755	. . . . 5 |- ( ( (/) e. { (/) } /\ C e. A ) -> <. (/) , C >. e. ( { (/) } X. A ) )
6	4, 5	mpan 424	. . . 4 |- ( C e. A -> <. (/) , C >. e. ( { (/) } X. A ) )
7		opeq2 3861	. . . . 5 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
8		df-inl 7237	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
9	7, 8	fvmptg 5718	. . . 4 |- ( ( C e. _V /\ <. (/) , C >. e. ( { (/) } X. A ) ) -> (inl ` C ) = <. (/) , C >. )
10	2, 6, 9	syl2anc 411	. . 3 |- ( C e. A -> (inl ` C ) = <. (/) , C >. )
11		elun1 3372	. . . . 5 |- ( <. (/) , C >. e. ( { (/) } X. A ) -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
12	6, 11	syl 14	. . . 4 |- ( C e. A -> <. (/) , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
13		df-dju 7228	. . . 4 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
14	12, 13	eleqtrrdi 2323	. . 3 |- ( C e. A -> <. (/) , C >. e. ( A ⊔ B ) )
15	10, 14	eqeltrd 2306	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
16	1, 15	eqeltrd 2306	1 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2800   u. cun 3196   (/)c0 3492   {csn 3667   <.cop 3670   X. cxp 4721   |` cres 4725   ` cfv 5324   1oc1o 6570   ⊔ cdju 7227  inlcinl 7235

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 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-dju 7228  df-inl 7237

This theorem is referenced by:  inlresf1  7251