Theorem releq 5114

Description: Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
releq	⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))

Proof of Theorem releq

Step	Hyp	Ref	Expression
1		sseq1 3589	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ (V × V) ↔ 𝐵 ⊆ (V × V)))
2		df-rel 5035	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
3		df-rel 5035	. 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
4	1, 2, 3	3bitr4g 302	1 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  Vcvv 3173   ⊆ wss 3540   × cxp 5026  Rel wrel 5033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-rel 5035

This theorem is referenced by:  releqi  5115  releqd  5116  dfrel2  5488  tposfn2  7239  ereq1  7614  isps  16974  isdir  17004  fpwrelmapffslem  28689  bnj1321  30143  frrlem6  30827  prtlem12  32964  relintabex  36700  clrellem  36742  clcnvlem  36743