Theorem pets2eq 40100

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 40099	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 39606	. . 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 40093	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 40094	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2796	1 ⊢ Pet2Parts = Pet2Ers

This proof depends on syntax axioms:   = wceq 1570   SucMap csucmap 39301   ShiftStable cshiftstable 39305   PetErs cpeters 39333   Pet2Ers cpet2ers 39334   PetParts cpetparts 39350   Pet2Parts cpet2parts 39351

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fo 6561  df-fv 6563  df-1st 8022  df-2nd 8023  df-ec 8744  df-qs 8748  df-xrn 39503  df-rels 39563  df-shiftstable 39605  df-coss 39624  df-coels 39625  df-ssr 39701  df-refs 39713  df-refrels 39714  df-refrel 39715  df-cnvrefs 39728  df-cnvrefrels 39729  df-cnvrefrel 39730  df-syms 39745  df-symrels 39746  df-symrel 39747  df-trs 39779  df-trrels 39780  df-trrel 39781  df-eqvrels 39791  df-eqvrel 39792  df-coeleqvrel 39794  df-dmqss 39845  df-dmqs 39846  df-ers 39871  df-erALTV 39872  df-comembers 39873  df-comember 39874  df-funALTV 39890  df-disjss 39911  df-disjs 39912  df-disjALTV 39913  df-eldisj 39915  df-parts 39991  df-part 39992  df-membparts 39993  df-membpart 39994  df-petparts 40091  df-peters 40092  df-pet2parts 40093  df-pet2ers 40094

This theorem is used by: (None)