Theorem resdmb 38189

Description: Biconditional form of resdm 6031, to replace it. (Contributed by BJ, 8-Nov-2026.) TODO: this is dfrel5 39537: move to main section and update credits.

Assertion

Ref	Expression
resdmb	⊢ (Rel 𝐴 ↔ (𝐴 ↾ dom 𝐴) = 𝐴)

Proof of Theorem resdmb

Step	Hyp	Ref	Expression
1		ssid 3935	. . 3 ⊢ dom 𝐴 ⊆ dom 𝐴
2	1	biantru 539	. 2 ⊢ (Rel 𝐴 ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴))
3		relssresb 38188	. 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) ↔ (𝐴 ↾ dom 𝐴) = 𝐴)
4	2, 3	bitri 278	1 ⊢ (Rel 𝐴 ↔ (𝐴 ↾ dom 𝐴) = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ⊆ wss 3900  dom cdm 5656   ↾ cres 5658  Rel wrel 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-dm 5666  df-rn 5667  df-res 5668

This theorem is used by:  bj-cnvresrn  38190