Theorem partimcomember 38811

Description: Partition with general 𝑅 (in addition to the member partition cf. mpet 38815 and mpet2 38816) 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 38784	. 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
2		mainer 38810	. 2 ⊢ ( ≀ 𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
3	1, 2	syl 17	1 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)

Syntax hints:   → wi 4   ≀ ccoss 38157   ErALTV werALTV 38183   CoMembEr wcomember 38185   Part wpart 38196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-eprel 5564  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ec 8729  df-qs 8733  df-coss 38387  df-coels 38388  df-refrel 38488  df-cnvrefrel 38503  df-symrel 38520  df-trrel 38550  df-eqvrel 38561  df-coeleqvrel 38563  df-dmqs 38615  df-erALTV 38640  df-comember 38642  df-funALTV 38658  df-disjALTV 38681  df-eldisj 38683  df-part 38742

This theorem is referenced by:  mainpart  38819