Theorem pets2eq 39829

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 39828	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 39335	. . 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 39822	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39823	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2793	1 ⊢ Pet2Parts = Pet2Ers

This proof depends on syntax axioms:   = wceq 1570   SucMap csucmap 39030   ShiftStable cshiftstable 39034   PetErs cpeters 39062   Pet2Ers cpet2ers 39063   PetParts cpetparts 39079   Pet2Parts cpet2parts 39080

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fo 6533  df-fv 6535  df-1st 7984  df-2nd 7985  df-ec 8697  df-qs 8701  df-xrn 39232  df-rels 39292  df-shiftstable 39334  df-coss 39353  df-coels 39354  df-ssr 39430  df-refs 39442  df-refrels 39443  df-refrel 39444  df-cnvrefs 39457  df-cnvrefrels 39458  df-cnvrefrel 39459  df-syms 39474  df-symrels 39475  df-symrel 39476  df-trs 39508  df-trrels 39509  df-trrel 39510  df-eqvrels 39520  df-eqvrel 39521  df-coeleqvrel 39523  df-dmqss 39574  df-dmqs 39575  df-ers 39600  df-erALTV 39601  df-comembers 39602  df-comember 39603  df-funALTV 39619  df-disjss 39640  df-disjs 39641  df-disjALTV 39642  df-eldisj 39644  df-parts 39720  df-part 39721  df-membparts 39722  df-membpart 39723  df-petparts 39820  df-peters 39821  df-pet2parts 39822  df-pet2ers 39823

This theorem is used by: (None)