Theorem mpet2 38312

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 38311 mpet3 38308, mostly in its conventional cpet 38310 and cpet2 38309 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38322 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 38311	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 38240	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 38138	. 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 205   E cep 5581  ^◡ccnv 5677   ↾ cres 5680   ≀ ccoss 37648   ErALTV werALTV 37674   CoMembEr wcomember 37676   Part wpart 37687   MembPart wmembpart 37689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-eprel 5582  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8726  df-qs 8730  df-coss 37883  df-coels 37884  df-refrel 37984  df-cnvrefrel 37999  df-symrel 38016  df-trrel 38046  df-eqvrel 38057  df-coeleqvrel 38059  df-dmqs 38111  df-erALTV 38136  df-comember 38138  df-funALTV 38154  df-disjALTV 38177  df-eldisj 38179  df-part 38238  df-membpart 38240

This theorem is referenced by:  mpets2  38313