Theorem mpet2 39199

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet 39198 mpet3 39195, mostly in its conventional cpet 39197 and cpet2 39196 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39209 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 39198	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
2		df-membpart 39116	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
3		df-comember 38996	. 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 206   E cep 5531  ^◡ccnv 5631   ↾ cres 5634   ≀ ccoss 38428   ErALTV werALTV 38454   CoMembEr wcomember 38458   Part wpart 38469   MembPart wmembpart 38471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8647  df-qs 8651  df-coss 38746  df-coels 38747  df-refrel 38837  df-cnvrefrel 38852  df-symrel 38869  df-trrel 38903  df-eqvrel 38914  df-coeleqvrel 38916  df-dmqs 38968  df-erALTV 38994  df-comember 38996  df-funALTV 39012  df-disjALTV 39035  df-eldisj 39037  df-part 39114  df-membpart 39116

This theorem is referenced by:  mpets2  39200