Theorem mpet2 38819

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 38818 mpet3 38815, mostly in its conventional cpet 38817 and cpet2 38816 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38829 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.)

Assertion

Ref	Expression
mpet2	⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)

Proof of Theorem mpet2

Step	Hyp	Ref	Expression
1		mpet 38818	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 38747	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 38645	. 2 ⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
4	1, 2, 3	3bitr3i 301	1 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)

Syntax hints:   ↔ wb 206   E cep 5581  ^◡ccnv 5682   ↾ cres 5685   ≀ ccoss 38160   ErALTV werALTV 38186   CoMembEr wcomember 38188   Part wpart 38199   MembPart wmembpart 38201

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  df-membpart 38747

This theorem is referenced by:  mpets2  38820