Theorem mpet2 38213

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 38212 mpet3 38209, mostly in its conventional cpet 38211 and cpet2 38210 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38223 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 38212	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 38141	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 38039	. 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 205   E cep 5570  ^◡ccnv 5666   ↾ cres 5669   ≀ ccoss 37546   ErALTV werALTV 37572   CoMembEr wcomember 37574   Part wpart 37585   MembPart wmembpart 37587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-eprel 5571  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702  df-qs 8706  df-coss 37784  df-coels 37785  df-refrel 37885  df-cnvrefrel 37900  df-symrel 37917  df-trrel 37947  df-eqvrel 37958  df-coeleqvrel 37960  df-dmqs 38012  df-erALTV 38037  df-comember 38039  df-funALTV 38055  df-disjALTV 38078  df-eldisj 38080  df-part 38139  df-membpart 38141

This theorem is referenced by:  mpets2  38214