Theorem mpet2 39328

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 39327 mpet3 39324, mostly in its conventional cpet 39326 and cpet2 39325 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39338 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 39327	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 39245	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 39125	. 2 ⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
4	1, 2, 3	3bitr3i 302	1 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)

Syntax hints:   ↔ wb 207   E cep 5524  ^◡ccnv 5624   ↾ cres 5627   ≀ ccoss 38557   ErALTV werALTV 38583   CoMembEr wcomember 38587   Part wpart 38598   MembPart wmembpart 38600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8642  df-qs 8646  df-coss 38875  df-coels 38876  df-refrel 38966  df-cnvrefrel 38981  df-symrel 38998  df-trrel 39032  df-eqvrel 39043  df-coeleqvrel 39045  df-dmqs 39097  df-erALTV 39123  df-comember 39125  df-funALTV 39141  df-disjALTV 39164  df-eldisj 39166  df-part 39243  df-membpart 39245

This theorem is referenced by:  mpets2  39329