Theorem mpet2 39886

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 39885 mpet3 39882, mostly in its conventional cpet 39884 and cpet2 39883 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39896 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 39885	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 39803	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 39683	. 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 5550  ^◡ccnv 5650   ↾ cres 5653   ≀ ccoss 39115   ErALTV werALTV 39141   CoMembEr wcomember 39145   Part wpart 39156   MembPart wmembpart 39158

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 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8719  df-qs 8723  df-coss 39433  df-coels 39434  df-refrel 39524  df-cnvrefrel 39539  df-symrel 39556  df-trrel 39590  df-eqvrel 39601  df-coeleqvrel 39603  df-dmqs 39655  df-erALTV 39681  df-comember 39683  df-funALTV 39699  df-disjALTV 39722  df-eldisj 39724  df-part 39801  df-membpart 39803

This theorem is used by:  mpets2  39887