Theorem djulclALT 12997

Description: Shortening of djulcl 6929 using djucllem 12996. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
djulclALT	|- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Proof of Theorem djulclALT

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4050	. . . . 5 |- (/) e. _V
2		df-inl 6925	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
3	1, 2	djucllem 12996	. . . 4 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( { (/) } X. A ) )
4	3	orcd 722	. . 3 |- ( C e. A -> ( ( (inl |` A ) ` C ) e. ( { (/) } X. A ) \/ ( (inl |` A ) ` C ) e. ( { 1o } X. B ) ) )
5		elun 3212	. . 3 |- ( ( (inl |` A ) ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) <-> ( ( (inl |` A ) ` C ) e. ( { (/) } X. A ) \/ ( (inl |` A ) ` C ) e. ( { 1o } X. B ) ) )
6	4, 5	sylibr 133	. 2 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
7		df-dju 6916	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
8	6, 7	eleqtrrdi 2231	1 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   \/ wo 697   e. wcel 1480   u. cun 3064   (/)c0 3358   {csn 3522   X. cxp 4532   |` cres 4536   ` cfv 5118   1oc1o 6299   ⊔ cdju 6915  inlcinl 6923

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-res 4546  df-iota 5083  df-fun 5120  df-fv 5126  df-dju 6916  df-inl 6925

This theorem is referenced by: (None)