Theorem partimcomember 37693

Description: Partition with general 𝑅 (in addition to the member partition cf. mpet 37697 and mpet2 37698) 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 37666	. 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
2		mainer 37692	. 2 ⊢ ( ≀ 𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
3	1, 2	syl 17	1 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)

Syntax hints:   → wi 4   ≀ ccoss 37031   ErALTV werALTV 37057   CoMembEr wcomember 37059   Part wpart 37070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-eprel 5579  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ec 8701  df-qs 8705  df-coss 37269  df-coels 37270  df-refrel 37370  df-cnvrefrel 37385  df-symrel 37402  df-trrel 37432  df-eqvrel 37443  df-coeleqvrel 37445  df-dmqs 37497  df-erALTV 37522  df-comember 37524  df-funALTV 37540  df-disjALTV 37563  df-eldisj 37565  df-part 37624

This theorem is referenced by:  mainpart  37701