Theorem pets2eq 39359

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 39358	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 38865	. . 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 39352	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39353	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2774	1 ⊢ Pet2Parts = Pet2Ers

Syntax hints:   = wceq 1548   SucMap csucmap 38560   ShiftStable cshiftstable 38564   PetErs cpeters 38592   Pet2Ers cpet2ers 38593   PetParts cpetparts 38609   Pet2Parts cpet2parts 38610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fo 6495  df-fv 6497  df-1st 7935  df-2nd 7936  df-ec 8639  df-qs 8643  df-xrn 38762  df-rels 38822  df-shiftstable 38864  df-coss 38883  df-coels 38884  df-ssr 38960  df-refs 38972  df-refrels 38973  df-refrel 38974  df-cnvrefs 38987  df-cnvrefrels 38988  df-cnvrefrel 38989  df-syms 39004  df-symrels 39005  df-symrel 39006  df-trs 39038  df-trrels 39039  df-trrel 39040  df-eqvrels 39050  df-eqvrel 39051  df-coeleqvrel 39053  df-dmqss 39104  df-dmqs 39105  df-ers 39130  df-erALTV 39131  df-comembers 39132  df-comember 39133  df-funALTV 39149  df-disjss 39170  df-disjs 39171  df-disjALTV 39172  df-eldisj 39174  df-parts 39250  df-part 39251  df-membparts 39252  df-membpart 39253  df-petparts 39350  df-peters 39351  df-pet2parts 39352  df-pet2ers 39353

This theorem is referenced by: (None)