Theorem partimcomember 38814

Description: Partition with general 𝑅 (in addition to the member partition cf. mpet 38818 and mpet2 38819) implies equivalent comembers. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)

Assertion

Ref	Expression
partimcomember	⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)

Proof of Theorem partimcomember

Step	Hyp	Ref	Expression
1		partim 38787	. 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
2		mainer 38813	. 2 ⊢ ( ≀ 𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
3	1, 2	syl 17	1 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)

Syntax hints:   → wi 4   ≀ ccoss 38160   ErALTV werALTV 38186   CoMembEr wcomember 38188   Part wpart 38199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-id 5576  df-eprel 5582  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-ec 8743  df-qs 8747  df-coss 38390  df-coels 38391  df-refrel 38491  df-cnvrefrel 38506  df-symrel 38523  df-trrel 38553  df-eqvrel 38564  df-coeleqvrel 38566  df-dmqs 38618  df-erALTV 38643  df-comember 38645  df-funALTV 38661  df-disjALTV 38684  df-eldisj 38686  df-part 38745

This theorem is referenced by:  mainpart  38822