Theorem djurclr 7213

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

Assertion

Ref	Expression
djurclr	|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )

Proof of Theorem djurclr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvres 5650	. 2 |- ( C e. B -> ( (inr |` B ) ` C ) = (inr ` C ) )
2		elex 2811	. . . 4 |- ( C e. B -> C e. _V )
3		1oex 6568	. . . . . 6 |- 1o e. _V
4	3	snid 3697	. . . . 5 |- 1o e. { 1o }
5		opelxpi 4750	. . . . 5 |- ( ( 1o e. { 1o } /\ C e. B ) -> <. 1o , C >. e. ( { 1o } X. B ) )
6	4, 5	mpan 424	. . . 4 |- ( C e. B -> <. 1o , C >. e. ( { 1o } X. B ) )
7		opeq2 3857	. . . . 5 |- ( x = C -> <. 1o , x >. = <. 1o , C >. )
8		df-inr 7211	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
9	7, 8	fvmptg 5709	. . . 4 |- ( ( C e. _V /\ <. 1o , C >. e. ( { 1o } X. B ) ) -> (inr ` C ) = <. 1o , C >. )
10	2, 6, 9	syl2anc 411	. . 3 |- ( C e. B -> (inr ` C ) = <. 1o , C >. )
11		elun2 3372	. . . . 5 |- ( <. 1o , C >. e. ( { 1o } X. B ) -> <. 1o , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
12	6, 11	syl 14	. . . 4 |- ( C e. B -> <. 1o , C >. e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
13		df-dju 7201	. . . 4 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
14	12, 13	eleqtrrdi 2323	. . 3 |- ( C e. B -> <. 1o , C >. e. ( A ⊔ B ) )
15	10, 14	eqeltrd 2306	. 2 |- ( C e. B -> (inr ` C ) e. ( A ⊔ B ) )
16	1, 15	eqeltrd 2306	1 |- ( C e. B -> ( (inr |` B ) ` 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 4716   |` cres 4720   ` cfv 5317   1oc1o 6553   ⊔ cdju 7200  inrcinr 7209

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-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

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 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-res 4730  df-iota 5277  df-fun 5319  df-fv 5325  df-1o 6560  df-dju 7201  df-inr 7211

This theorem is referenced by:  inrresf1  7225