Theorem mpet2 39663

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 39662 mpet3 39659, mostly in its conventional cpet 39661 and cpet2 39660 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39673 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 39662	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 39580	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 39460	. 2 ⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
4	1, 2, 3	3bitr3i 304	1 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   E cep 5562  ^◡ccnv 5662   ↾ cres 5665   ≀ ccoss 38892   ErALTV werALTV 38918   CoMembEr wcomember 38922   Part wpart 38933   MembPart wmembpart 38935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8702  df-qs 8706  df-coss 39210  df-coels 39211  df-refrel 39301  df-cnvrefrel 39316  df-symrel 39333  df-trrel 39367  df-eqvrel 39378  df-coeleqvrel 39380  df-dmqs 39432  df-erALTV 39458  df-comember 39460  df-funALTV 39476  df-disjALTV 39499  df-eldisj 39501  df-part 39578  df-membpart 39580

This theorem is used by:  mpets2  39664