Theorem djulclALT 14693

Description: Shortening of djulcl 7053 using djucllem 14692. (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 4132	. . . . 5 |- (/) e. _V
2		df-inl 7049	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
3	1, 2	djucllem 14692	. . . 4 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( { (/) } X. A ) )
4	3	orcd 733	. . 3 |- ( C e. A -> ( ( (inl |` A ) ` C ) e. ( { (/) } X. A ) \/ ( (inl |` A ) ` C ) e. ( { 1o } X. B ) ) )
5		elun 3278	. . 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 134	. 2 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
7		df-dju 7040	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
8	6, 7	eleqtrrdi 2271	1 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   \/ wo 708   e. wcel 2148   u. cun 3129   (/)c0 3424   {csn 3594   X. cxp 4626   |` cres 4630   ` cfv 5218   1oc1o 6413   ⊔ cdju 7039  inlcinl 7047

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-dju 7040  df-inl 7049

This theorem is referenced by: (None)