Theorem pets2eq 39910

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 39909	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 39416	. . 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 39903	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39904	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2793	1 ⊢ Pet2Parts = Pet2Ers

This proof depends on syntax axioms:   = wceq 1569   SucMap csucmap 39111   ShiftStable cshiftstable 39115   PetErs cpeters 39143   Pet2Ers cpet2ers 39144   PetParts cpetparts 39160   Pet2Parts cpet2parts 39161

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 8001  df-2nd 8002  df-ec 8719  df-qs 8723  df-xrn 39313  df-rels 39373  df-shiftstable 39415  df-coss 39434  df-coels 39435  df-ssr 39511  df-refs 39523  df-refrels 39524  df-refrel 39525  df-cnvrefs 39538  df-cnvrefrels 39539  df-cnvrefrel 39540  df-syms 39555  df-symrels 39556  df-symrel 39557  df-trs 39589  df-trrels 39590  df-trrel 39591  df-eqvrels 39601  df-eqvrel 39602  df-coeleqvrel 39604  df-dmqss 39655  df-dmqs 39656  df-ers 39681  df-erALTV 39682  df-comembers 39683  df-comember 39684  df-funALTV 39700  df-disjss 39721  df-disjs 39722  df-disjALTV 39723  df-eldisj 39725  df-parts 39801  df-part 39802  df-membparts 39803  df-membpart 39804  df-petparts 39901  df-peters 39902  df-pet2parts 39903  df-pet2ers 39904

This theorem is used by: (None)