Theorem djulclALT 15937

Description: Shortening of djulcl 7179 using djucllem 15936. (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 4187	. . . . 5 |- (/) e. _V
2		df-inl 7175	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
3	1, 2	djucllem 15936	. . . 4 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( { (/) } X. A ) )
4	3	orcd 735	. . 3 |- ( C e. A -> ( ( (inl |` A ) ` C ) e. ( { (/) } X. A ) \/ ( (inl |` A ) ` C ) e. ( { 1o } X. B ) ) )
5		elun 3322	. . 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 7166	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
8	6, 7	eleqtrrdi 2301	1 |- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   \/ wo 710   e. wcel 2178   u. cun 3172   (/)c0 3468   {csn 3643   X. cxp 4691   |` cres 4695   ` cfv 5290   1oc1o 6518   ⊔ cdju 7165  inlcinl 7173

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-res 4705  df-iota 5251  df-fun 5292  df-fv 5298  df-dju 7166  df-inl 7175

This theorem is referenced by: (None)