Theorem mpet2 39703

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 39702 mpet3 39699, mostly in its conventional cpet 39701 and cpet2 39700 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39713 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 39702	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 39620	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 39500	. 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 5554  ^◡ccnv 5654   ↾ cres 5657   ≀ ccoss 38932   ErALTV werALTV 38958   CoMembEr wcomember 38962   Part wpart 38973   MembPart wmembpart 38975

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8699  df-qs 8703  df-coss 39250  df-coels 39251  df-refrel 39341  df-cnvrefrel 39356  df-symrel 39373  df-trrel 39407  df-eqvrel 39418  df-coeleqvrel 39420  df-dmqs 39472  df-erALTV 39498  df-comember 39500  df-funALTV 39516  df-disjALTV 39539  df-eldisj 39541  df-part 39618  df-membpart 39620

This theorem is used by:  mpets2  39704