Theorem pets2eq 39481

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 39480	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 38987	. . 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 39474	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39475	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2797	1 ⊢ Pet2Parts = Pet2Ers

Syntax hints:   = wceq 1562   SucMap csucmap 38682   ShiftStable cshiftstable 38686   PetErs cpeters 38714   Pet2Ers cpet2ers 38715   PetParts cpetparts 38731   Pet2Parts cpet2parts 38732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-eprel 5549  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fo 6529  df-fv 6531  df-1st 7972  df-2nd 7973  df-ec 8682  df-qs 8686  df-xrn 38884  df-rels 38944  df-shiftstable 38986  df-coss 39005  df-coels 39006  df-ssr 39082  df-refs 39094  df-refrels 39095  df-refrel 39096  df-cnvrefs 39109  df-cnvrefrels 39110  df-cnvrefrel 39111  df-syms 39126  df-symrels 39127  df-symrel 39128  df-trs 39160  df-trrels 39161  df-trrel 39162  df-eqvrels 39172  df-eqvrel 39173  df-coeleqvrel 39175  df-dmqss 39226  df-dmqs 39227  df-ers 39252  df-erALTV 39253  df-comembers 39254  df-comember 39255  df-funALTV 39271  df-disjss 39292  df-disjs 39293  df-disjALTV 39294  df-eldisj 39296  df-parts 39372  df-part 39373  df-membparts 39374  df-membpart 39375  df-petparts 39472  df-peters 39473  df-pet2parts 39474  df-pet2ers 39475

This theorem is referenced by: (None)