Theorem djulclr 7247

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 5663	. 2 |- ( C e. A -> ( (inl |` A ) ` C ) = (inl ` C ) )
2		elex 2814	. . . 4 |- ( C e. A -> C e. _V )
3		0ex 4216	. . . . . 6 |- (/) e. _V
4	3	snid 3700	. . . . 5 |- (/) e. { (/) }
5		opelxpi 4757	. . . . 5 |- ( ( (/) e. { (/) } /\ C e. A ) -> <. (/) , C >. e. ( { (/) } X. A ) )
6	4, 5	mpan 424	. . . 4 |- ( C e. A -> <. (/) , C >. e. ( { (/) } X. A ) )
7		opeq2 3863	. . . . 5 |- ( x = C -> <. (/) , x >. = <. (/) , C >. )
8		df-inl 7245	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
9	7, 8	fvmptg 5722	. . . 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 3374	. . . . 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 7236	. . . 4 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
14	12, 13	eleqtrrdi 2325	. . 3 |- ( C e. A -> <. (/) , C >. e. ( A ⊔ B ) )
15	10, 14	eqeltrd 2308	. 2 |- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
16	1, 15	eqeltrd 2308	1 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   u. cun 3198   (/)c0 3494   {csn 3669   <.cop 3672   X. cxp 4723   |` cres 4727   ` cfv 5326   1oc1o 6574   ⊔ cdju 7235  inlcinl 7243

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-dju 7236  df-inl 7245

This theorem is referenced by:  inlresf1  7259