Theorem djulclr 7026

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 5520	. 2 |- ( C e. A -> ( (inl |` A ) ` C ) = (inl ` C ) )
2		elex 2741	. . . 4 |- ( C e. A -> C e. _V )
3		0ex 4116	. . . . . 6 |- (/) e. _V
4	3	snid 3614	. . . . 5 |- (/) e. { (/) }
5		opelxpi 4643	. . . . 5 |- ( ( (/) e. { (/) } /\ C e. A ) -> <. (/) , C >. e. ( { (/) } X. A ) )
6	4, 5	mpan 422	. . . 4 |- ( C e. A -> <. (/) , C >. e. ( { (/) } X. A ) )
7		opeq2 3766	. . . . 5 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
8		df-inl 7024	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
9	7, 8	fvmptg 5572	. . . 4 |- ( ( C e. _V /\ <. (/) , C >. e. ( { (/) } X. A ) ) -> (inl ` C ) = <. (/) , C >. )
10	2, 6, 9	syl2anc 409	. . 3 |- ( C e. A -> (inl ` C ) = <. (/) , C >. )
11		elun1 3294	. . . . 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 7015	. . . 4 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
14	12, 13	eleqtrrdi 2264	. . 3 |- ( C e. A -> <. (/) , C >. e. ( A ⊔ B ) )
15	10, 14	eqeltrd 2247	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
16	1, 15	eqeltrd 2247	1 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   _Vcvv 2730   u. cun 3119   (/)c0 3414   {csn 3583   <.cop 3586   X. cxp 4609   |` cres 4613   ` cfv 5198   1oc1o 6388   ⊔ cdju 7014  inlcinl 7022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fv 5206  df-dju 7015  df-inl 7024

This theorem is referenced by:  inlresf1  7038