Theorem pets2eq 39318

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 39317	. . 3 ⊢ PetParts = PetErs
2		shiftstableeq2 38824	. . 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 39311	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
5		df-pet2ers 39312	. 2 ⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
6	3, 4, 5	3eqtr4i 2770	1 ⊢ Pet2Parts = Pet2Ers

Syntax hints:   = wceq 1542   SucMap csucmap 38519   ShiftStable cshiftstable 38523   PetErs cpeters 38551   Pet2Ers cpet2ers 38552   PetParts cpetparts 38568   Pet2Parts cpet2parts 38569

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

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 3343  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-eprel 5526  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-fo 6500  df-fv 6502  df-1st 7937  df-2nd 7938  df-ec 8640  df-qs 8644  df-xrn 38721  df-rels 38781  df-shiftstable 38823  df-coss 38842  df-coels 38843  df-ssr 38919  df-refs 38931  df-refrels 38932  df-refrel 38933  df-cnvrefs 38946  df-cnvrefrels 38947  df-cnvrefrel 38948  df-syms 38963  df-symrels 38964  df-symrel 38965  df-trs 38997  df-trrels 38998  df-trrel 38999  df-eqvrels 39009  df-eqvrel 39010  df-coeleqvrel 39012  df-dmqss 39063  df-dmqs 39064  df-ers 39089  df-erALTV 39090  df-comembers 39091  df-comember 39092  df-funALTV 39108  df-disjss 39129  df-disjs 39130  df-disjALTV 39131  df-eldisj 39133  df-parts 39209  df-part 39210  df-membparts 39211  df-membpart 39212  df-petparts 39309  df-peters 39310  df-pet2parts 39311  df-pet2ers 39312

This theorem is referenced by: (None)