Theorem djulclr 7115

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 5582	. 2 |- ( C e. A -> ( (inl |` A ) ` C ) = (inl ` C ) )
2		elex 2774	. . . 4 |- ( C e. A -> C e. _V )
3		0ex 4160	. . . . . 6 |- (/) e. _V
4	3	snid 3653	. . . . 5 |- (/) e. { (/) }
5		opelxpi 4695	. . . . 5 |- ( ( (/) e. { (/) } /\ C e. A ) -> <. (/) , C >. e. ( { (/) } X. A ) )
6	4, 5	mpan 424	. . . 4 |- ( C e. A -> <. (/) , C >. e. ( { (/) } X. A ) )
7		opeq2 3809	. . . . 5 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
8		df-inl 7113	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
9	7, 8	fvmptg 5637	. . . 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 3330	. . . . 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 7104	. . . 4 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
14	12, 13	eleqtrrdi 2290	. . 3 |- ( C e. A -> <. (/) , C >. e. ( A ⊔ B ) )
15	10, 14	eqeltrd 2273	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
16	1, 15	eqeltrd 2273	1 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   u. cun 3155   (/)c0 3450   {csn 3622   <.cop 3625   X. cxp 4661   |` cres 4665   ` cfv 5258   1oc1o 6467   ⊔ cdju 7103  inlcinl 7111

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-res 4675  df-iota 5219  df-fun 5260  df-fv 5266  df-dju 7104  df-inl 7113

This theorem is referenced by:  inlresf1  7127