Theorem partimcomember 38783

Description: Partition with general 𝑅 (in addition to the member partition cf. mpet 38787 and mpet2 38788) 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 38756	. 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
2		mainer 38782	. 2 ⊢ ( ≀ 𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
3	1, 2	syl 17	1 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)

Syntax hints:   → wi 4   ≀ ccoss 38127   ErALTV werALTV 38153   CoMembEr wcomember 38155   Part wpart 38166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-eprel 5599  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-ec 8759  df-qs 8763  df-coss 38359  df-coels 38360  df-refrel 38460  df-cnvrefrel 38475  df-symrel 38492  df-trrel 38522  df-eqvrel 38533  df-coeleqvrel 38535  df-dmqs 38587  df-erALTV 38612  df-comember 38614  df-funALTV 38630  df-disjALTV 38653  df-eldisj 38655  df-part 38714

This theorem is referenced by:  mainpart  38791