Theorem pets2eq 39180

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 39179	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 38686	. . 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 39173	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39174	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2770	1 ⊢ Pet2Parts = Pet2Ers

Syntax hints:   = wceq 1542   SucMap csucmap 38381   ShiftStable cshiftstable 38385   PetErs cpeters 38413   Pet2Ers cpet2ers 38414   PetParts cpetparts 38430   Pet2Parts cpet2parts 38431

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-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 3062  df-rmo 3351  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7935  df-2nd 7936  df-ec 8639  df-qs 8643  df-xrn 38583  df-rels 38643  df-shiftstable 38685  df-coss 38704  df-coels 38705  df-ssr 38781  df-refs 38793  df-refrels 38794  df-refrel 38795  df-cnvrefs 38808  df-cnvrefrels 38809  df-cnvrefrel 38810  df-syms 38825  df-symrels 38826  df-symrel 38827  df-trs 38859  df-trrels 38860  df-trrel 38861  df-eqvrels 38871  df-eqvrel 38872  df-coeleqvrel 38874  df-dmqss 38925  df-dmqs 38926  df-ers 38951  df-erALTV 38952  df-comembers 38953  df-comember 38954  df-funALTV 38970  df-disjss 38991  df-disjs 38992  df-disjALTV 38993  df-eldisj 38995  df-parts 39071  df-part 39072  df-membparts 39073  df-membpart 39074  df-petparts 39171  df-peters 39172  df-pet2parts 39173  df-pet2ers 39174

This theorem is referenced by: (None)