Theorem pets2eq 39298

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 39297	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 38804	. . 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 39291	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39292	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2769	1 ⊢ Pet2Parts = Pet2Ers

Syntax hints:   = wceq 1542   SucMap csucmap 38499   ShiftStable cshiftstable 38503   PetErs cpeters 38531   Pet2Ers cpet2ers 38532   PetParts cpetparts 38548   Pet2Parts cpet2parts 38549

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-1st 7942  df-2nd 7943  df-ec 8645  df-qs 8649  df-xrn 38701  df-rels 38761  df-shiftstable 38803  df-coss 38822  df-coels 38823  df-ssr 38899  df-refs 38911  df-refrels 38912  df-refrel 38913  df-cnvrefs 38926  df-cnvrefrels 38927  df-cnvrefrel 38928  df-syms 38943  df-symrels 38944  df-symrel 38945  df-trs 38977  df-trrels 38978  df-trrel 38979  df-eqvrels 38989  df-eqvrel 38990  df-coeleqvrel 38992  df-dmqss 39043  df-dmqs 39044  df-ers 39069  df-erALTV 39070  df-comembers 39071  df-comember 39072  df-funALTV 39088  df-disjss 39109  df-disjs 39110  df-disjALTV 39111  df-eldisj 39113  df-parts 39189  df-part 39190  df-membparts 39191  df-membpart 39192  df-petparts 39289  df-peters 39290  df-pet2parts 39291  df-pet2ers 39292

This theorem is referenced by: (None)