Theorem partimcomember 39881

Description: Partition with general 𝑅 (in addition to the member partition cf. mpet 39885 and mpet2 39886) 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 39843	. 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
2		mainer 39880	. 2 ⊢ ( ≀ 𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
3	1, 2	syl 18	1 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)

This proof depends on syntax axioms:   → wi 4   ≀ ccoss 39115   ErALTV werALTV 39141   CoMembEr wcomember 39145   Part wpart 39156

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8719  df-qs 8723  df-coss 39433  df-coels 39434  df-refrel 39524  df-cnvrefrel 39539  df-symrel 39556  df-trrel 39590  df-eqvrel 39601  df-coeleqvrel 39603  df-dmqs 39655  df-erALTV 39681  df-comember 39683  df-funALTV 39699  df-disjALTV 39722  df-eldisj 39724  df-part 39801

This theorem is used by:  mainpart  39889