Theorem djurclr 7380

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 5714	. 2 |- ( C e. B -> ( (inr |` B ) ` C ) = (inr ` C ) )
2		elex 2833	. . . 4 |- ( C e. B -> C e. _V )
3		1oex 6685	. . . . . 6 |- 1o e. _V
4	3	snid 3736	. . . . 5 |- 1o e. { 1o }
5		opelxpi 4801	. . . . 5 |- ( ( 1o e. { 1o } /\ C e. B ) -> <. 1o , C >. e. ( { 1o } X. B ) )
6	4, 5	mpan 428	. . . 4 |- ( C e. B -> <. 1o , C >. e. ( { 1o } X. B ) )
7		opeq2 3900	. . . . 5 |- ( x = C -> <. 1o , x >. = <. 1o , C >. )
8		df-inr 7378	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
9	7, 8	fvmptg 5775	. . . 4 |- ( ( C e. _V /\ <. 1o , C >. e. ( { 1o } X. B ) ) -> (inr ` C ) = <. 1o , C >. )
10	2, 6, 9	syl2anc 415	. . 3 |- ( C e. B -> (inr ` C ) = <. 1o , C >. )
11		elun2 3397	. . . . 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 7368	. . . 4 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
14	12, 13	eleqtrrdi 2332	. . 3 |- ( C e. B -> <. 1o , C >. e. ( A ⊔ B ) )
15	10, 14	eqeltrd 2315	. 2 |- ( C e. B -> (inr ` C ) e. ( A ⊔ B ) )
16	1, 15	eqeltrd 2315	1 |- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   (/)c0 3520   {csn 3705   <.cop 3708   X. cxp 4767   |` cres 4771   ` cfv 5372   1oc1o 6670   ⊔ cdju 7367  inrcinr 7376

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-1o 6677  df-dju 7368  df-inr 7378

This theorem is referenced by:  inrresf1  7392