Theorem releq 5761

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 3938	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ (V × V) ↔ 𝐵 ⊆ (V × V)))
2		df-rel 5665	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
3		df-rel 5665	. 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
4	1, 2, 3	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Vcvv 3454   ⊆ wss 3900   × cxp 5656  Rel wrel 5663

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-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ss 3901  df-rel 5665

This theorem is used by:  releqi  5762  releqd  5763  relsnb  5789  dfrel2  6199  cnveq0b  6210  tposfn2  8303  ereq1  8763  isps  18792  isdir  18822  fpwrelmapffslem  33420  bnj1321  35768  refreleq  39799  symreleq  39840  trreleq  39864  prtlem12  40190  relintabex  44843  clrellem  44883  clcnvlem  44884  rellan  51137  relran  51138