Theorem mpet2 38827

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 38826 mpet3 38823, mostly in its conventional cpet 38825 and cpet2 38824 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38837 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 38826	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 38755	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 38653	. 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 5539  ^◡ccnv 5639   ↾ cres 5642   ≀ ccoss 38164   ErALTV werALTV 38190   CoMembEr wcomember 38192   Part wpart 38203   MembPart wmembpart 38205

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-eprel 5540  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ec 8675  df-qs 8679  df-coss 38397  df-coels 38398  df-refrel 38498  df-cnvrefrel 38513  df-symrel 38530  df-trrel 38560  df-eqvrel 38571  df-coeleqvrel 38573  df-dmqs 38625  df-erALTV 38651  df-comember 38653  df-funALTV 38669  df-disjALTV 38692  df-eldisj 38694  df-part 38753  df-membpart 38755

This theorem is referenced by:  mpets2  38828