Theorem pets2eq 39257

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 39256	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 38763	. . 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 39250	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39251	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2770	1 ⊢ Pet2Parts = Pet2Ers

Syntax hints:   = wceq 1542   SucMap csucmap 38458   ShiftStable cshiftstable 38462   PetErs cpeters 38490   Pet2Ers cpet2ers 38491   PetParts cpetparts 38507   Pet2Parts cpet2parts 38508

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

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 3063  df-rmo 3352  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-eprel 5534  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fo 6508  df-fv 6510  df-1st 7945  df-2nd 7946  df-ec 8649  df-qs 8653  df-xrn 38660  df-rels 38720  df-shiftstable 38762  df-coss 38781  df-coels 38782  df-ssr 38858  df-refs 38870  df-refrels 38871  df-refrel 38872  df-cnvrefs 38885  df-cnvrefrels 38886  df-cnvrefrel 38887  df-syms 38902  df-symrels 38903  df-symrel 38904  df-trs 38936  df-trrels 38937  df-trrel 38938  df-eqvrels 38948  df-eqvrel 38949  df-coeleqvrel 38951  df-dmqss 39002  df-dmqs 39003  df-ers 39028  df-erALTV 39029  df-comembers 39030  df-comember 39031  df-funALTV 39047  df-disjss 39068  df-disjs 39069  df-disjALTV 39070  df-eldisj 39072  df-parts 39148  df-part 39149  df-membparts 39150  df-membpart 39151  df-petparts 39248  df-peters 39249  df-pet2parts 39250  df-pet2ers 39251

This theorem is referenced by: (None)