Theorem pet 38549

Description: Partition-Equivalence Theorem with general 𝑅 while preserving the restricted converse epsilon relation of mpet2 38538 (as opposed to petincnvepres 38547). 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 38150.

This theorem (together with pets 38550 and pet2 38548) is the main result of my investigation into set theory. It is no more general than the conventional Member Partition-Equivalence Theorem mpet 38537, mpet2 38538 and mpet3 38534 (because you cannot set 𝑅 in this theorem in such a way that you get mpet2 38538), 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 38538 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 38548	. 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 38467	. 2 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴))
3		dferALTV2 38366	. 2 ⊢ ( ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴))
4	1, 2, 3	3bitr4i 302	1 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴)

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1534   E cep 5585  ^◡ccnv 5681  dom cdm 5682   ↾ cres 5684   / cqs 8733   ⋉ cxrn 37875   ≀ ccoss 37876   EqvRel weqvrel 37893   ErALTV werALTV 37902   Disj wdisjALTV 37910   Part wpart 37915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-eprel 5586  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fo 6560  df-fv 6562  df-1st 8003  df-2nd 8004  df-ec 8736  df-qs 8740  df-xrn 38069  df-coss 38109  df-refrel 38210  df-cnvrefrel 38225  df-symrel 38242  df-trrel 38272  df-eqvrel 38283  df-dmqs 38337  df-erALTV 38362  df-funALTV 38380  df-disjALTV 38403  df-eldisj 38405  df-part 38464

This theorem is referenced by:  pets  38550