Theorem dfmember 36784

Description: Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)

Assertion

Ref	Expression
dfmember	⊢ ( MembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

Proof of Theorem dfmember

Step	Hyp	Ref	Expression
1		df-member 36778	. 2 ⊢ ( MembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
2		df-coels 36538	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	erALTVeq1i 36782	. 2 ⊢ ( ∼ 𝐴 ErALTV 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
4	1, 3	bitr4i 277	1 ⊢ ( MembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

Syntax hints:   ↔ wb 205   E cep 5494  ^◡ccnv 5588   ↾ cres 5591   ≀ ccoss 36333   ∼ ccoels 36334   ErALTV werALTV 36359   MembEr wmember 36361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-qs 8504  df-coels 36538  df-refrel 36630  df-symrel 36658  df-trrel 36688  df-eqvrel 36698  df-dmqs 36752  df-erALTV 36776  df-member 36778

This theorem is referenced by:  dfmember2  36785