Theorem pet 39345

Description: Partition-Equivalence Theorem with general 𝑅 while preserving the restricted converse epsilon relation of mpet2 39334 (as opposed to petincnvepres 39343). A class is a partition by a range Cartesian product with general 𝑅 and the restricted converse element class if and only if the cosets by the range Cartesian product are in an equivalence relation on it. Cf. br1cossxrncnvepres 38922.

This theorem (together with pets 39346 and pet2 39344) is the main result of my investigation into set theory. It is no more general than the conventional Member Partition-Equivalence Theorem mpet 39333, mpet2 39334 and mpet3 39330 (because you cannot set 𝑅 in this theorem in such a way that you get mpet2 39334), i.e., it is not the hypothetical General Partition-Equivalence Theorem gpet ⊢ (𝑅 Part 𝐴 ↔ ≀ 𝑅 ErALTV 𝐴), but this one has a general part that mpet2 39334 lacks: 𝑅, which is sufficient for my future application of set theory, for my purpose outside of set theory. (Contributed by Peter Mazsa, 23-Sep-2021.)

Assertion

Ref	Expression
pet	⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴)

Proof of Theorem pet

Step	Hyp	Ref	Expression
1		pet2 39344	. 2 ⊢ (( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴))
2		dfpart2 39252	. 2 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴))
3		dferALTV2 39133	. 2 ⊢ ( ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴))
4	1, 2, 3	3bitr4i 305	1 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴)

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548   E cep 5519  ^◡ccnv 5619  dom cdm 5620   ↾ cres 5622   / cqs 8636   ⋉ cxrn 38554   ≀ ccoss 38563   EqvRel weqvrel 38580   ErALTV werALTV 38589   Disj wdisjALTV 38599   Part wpart 38604

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-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681

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 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-eprel 5520  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fo 6494  df-fv 6496  df-1st 7933  df-2nd 7934  df-ec 8639  df-qs 8643  df-xrn 38760  df-coss 38881  df-refrel 38972  df-cnvrefrel 38987  df-symrel 39004  df-trrel 39038  df-eqvrel 39049  df-dmqs 39103  df-erALTV 39129  df-funALTV 39147  df-disjALTV 39170  df-eldisj 39172  df-part 39249

This theorem is referenced by:  pets  39346