Theorem djurclALT 16700

Description: Shortening of djurcl 7356 using djucllem 16698. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
djurclALT	|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )

Proof of Theorem djurclALT

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1oex 6668	. . . . 5 |- 1o e. _V
2		df-inr 7352	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
3	1, 2	djucllem 16698	. . . 4 |- ( C e. B -> ( (inr |` B ) ` C ) e. ( { 1o } X. B ) )
4	3	olcd 742	. . 3 |- ( C e. B -> ( ( (inr |` B ) ` C ) e. ( { (/) } X. A ) \/ ( (inr |` B ) ` C ) e. ( { 1o } X. B ) ) )
5		elun 3364	. . 3 |- ( ( (inr |` B ) ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) <-> ( ( (inr |` B ) ` C ) e. ( { (/) } X. A ) \/ ( (inr |` B ) ` C ) e. ( { 1o } X. B ) ) )
6	4, 5	sylibr 134	. 2 |- ( C e. B -> ( (inr |` B ) ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
7		df-dju 7342	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
8	6, 7	eleqtrrdi 2328	1 |- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   \/ wo 716   e. wcel 2205   u. cun 3212   (/)c0 3512   {csn 3694   X. cxp 4752   |` cres 4756   ` cfv 5357   1oc1o 6653   ⊔ cdju 7341  inrcinr 7350

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365  df-1o 6660  df-dju 7342  df-inr 7352

This theorem is referenced by: (None)