Theorem mpet2 39623

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 39622 mpet3 39619, mostly in its conventional cpet 39621 and cpet2 39620 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39633 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 39622	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 39540	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 39420	. 2 ⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
4	1, 2, 3	3bitr3i 304	1 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)

Syntax hints:   ↔ wb 209   E cep 5560  ^◡ccnv 5660   ↾ cres 5663   ≀ ccoss 38852   ErALTV werALTV 38878   CoMembEr wcomember 38882   Part wpart 38893   MembPart wmembpart 38895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-qs 8696  df-coss 39170  df-coels 39171  df-refrel 39261  df-cnvrefrel 39276  df-symrel 39293  df-trrel 39327  df-eqvrel 39338  df-coeleqvrel 39340  df-dmqs 39392  df-erALTV 39418  df-comember 39420  df-funALTV 39436  df-disjALTV 39459  df-eldisj 39461  df-part 39538  df-membpart 39540

This theorem is referenced by:  mpets2  39624