Theorem pets2eq 39576

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 39575	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 39082	. . 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 39569	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39570	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2803	1 ⊢ Pet2Parts = Pet2Ers

Syntax hints:   = wceq 1568   SucMap csucmap 38777   ShiftStable cshiftstable 38781   PetErs cpeters 38809   Pet2Ers cpet2ers 38810   PetParts cpetparts 38826   Pet2Parts cpet2parts 38827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-eprel 5565  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7989  df-2nd 7990  df-ec 8699  df-qs 8703  df-xrn 38979  df-rels 39039  df-shiftstable 39081  df-coss 39100  df-coels 39101  df-ssr 39177  df-refs 39189  df-refrels 39190  df-refrel 39191  df-cnvrefs 39204  df-cnvrefrels 39205  df-cnvrefrel 39206  df-syms 39221  df-symrels 39222  df-symrel 39223  df-trs 39255  df-trrels 39256  df-trrel 39257  df-eqvrels 39267  df-eqvrel 39268  df-coeleqvrel 39270  df-dmqss 39321  df-dmqs 39322  df-ers 39347  df-erALTV 39348  df-comembers 39349  df-comember 39350  df-funALTV 39366  df-disjss 39387  df-disjs 39388  df-disjALTV 39389  df-eldisj 39391  df-parts 39467  df-part 39468  df-membparts 39469  df-membpart 39470  df-petparts 39567  df-peters 39568  df-pet2parts 39569  df-pet2ers 39570

This theorem is referenced by: (None)