Theorem dfrel6 35157

Description: Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)

Assertion

Ref	Expression
dfrel6	⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)

Proof of Theorem dfrel6

Step	Hyp	Ref	Expression
1		dfrel5 35156	. 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
2		dfres3 5746	. . 3 ⊢ (𝑅 ↾ dom 𝑅) = (𝑅 ∩ (dom 𝑅 × ran 𝑅))
3	2	eqeq1i 2802	. 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
4	1, 3	bitri 276	1 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)

Syntax hints:   ↔ wb 207   = wceq 1525   ∩ cin 3864   × cxp 5448  dom cdm 5450  ran crn 5451   ↾ cres 5452  Rel wrel 5455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-xp 5456  df-rel 5457  df-cnv 5458  df-dm 5460  df-rn 5461  df-res 5462

This theorem is referenced by:  elrels6  35282  dfrefrel2  35307  dfcnvrefrel2  35320  dfsymrel2  35337  dftrrel2  35365