Theorem pets2eq 39654

Description: Grade-stable generalized partition-equivalence identification. After applying the same grade-stability operator (SucMap ShiftStable) to both sides, the grade-stable pet classes still coincide. Confirms that the grade/tower infrastructure is orthogonal to the partition-vs-equivalence viewpoint: stability is preserved under the PetParts = PetErs identification. This is the level at which we can freely work on whichever side is more convenient (Parts for block discipline, Ers for equivalence reasoning), without changing the stable notion of "pet". (Contributed by Peter Mazsa, 19-Feb-2026.)

Assertion

Ref	Expression
pets2eq	⊢ Pet2Parts = Pet2Ers

Proof of Theorem pets2eq

Step	Hyp	Ref	Expression
1		petseq 39653	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 39160	. . 3 ⊢ ( PetParts = PetErs → ( SucMap ShiftStable PetParts ) = ( SucMap ShiftStable PetErs ))
3	1, 2	ax-mp 5	. 2 ⊢ ( SucMap ShiftStable PetParts ) = ( SucMap ShiftStable PetErs )
4		df-pet2parts 39647	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39648	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2795	1 ⊢ Pet2Parts = Pet2Ers

This proof depends on syntax axioms:   = wceq 1569   SucMap csucmap 38855   ShiftStable cshiftstable 38859   PetErs cpeters 38887   Pet2Ers cpet2ers 38888   PetParts cpetparts 38904   Pet2Parts cpet2parts 38905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-eprel 5560  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7984  df-2nd 7985  df-ec 8694  df-qs 8698  df-xrn 39057  df-rels 39117  df-shiftstable 39159  df-coss 39178  df-coels 39179  df-ssr 39255  df-refs 39267  df-refrels 39268  df-refrel 39269  df-cnvrefs 39282  df-cnvrefrels 39283  df-cnvrefrel 39284  df-syms 39299  df-symrels 39300  df-symrel 39301  df-trs 39333  df-trrels 39334  df-trrel 39335  df-eqvrels 39345  df-eqvrel 39346  df-coeleqvrel 39348  df-dmqss 39399  df-dmqs 39400  df-ers 39425  df-erALTV 39426  df-comembers 39427  df-comember 39428  df-funALTV 39444  df-disjss 39465  df-disjs 39466  df-disjALTV 39467  df-eldisj 39469  df-parts 39545  df-part 39546  df-membparts 39547  df-membpart 39548  df-petparts 39645  df-peters 39646  df-pet2parts 39647  df-pet2ers 39648

This theorem is used by: (None)