Theorem hbrel 2478

Description: Bound-variable hypothesis builder for a relation.

Hypothesis

Ref	Expression
hbrel.1	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
hbrel	⊢ (Rel A → ∀xRel A)

Distinct variable group(s):   y,A   x,y

Proof of Theorem hbrel

Step	Hyp	Ref	Expression
1		hbrel.1	. . 3 ⊢ (y ∈ A → ∀x y ∈ A)
2		ax-17 925	. . 3 ⊢ (y ∈ (V × V) → ∀x y ∈ (V × V))
3	1, 2	hbss 1501	. 2 ⊢ (A ⊆ (V × V) → ∀x A ⊆ (V × V))
4		df-rel 2425	. 2 ⊢ (Rel A ↔ A ⊆ (V × V))
5	4	bial 695	. 2 ⊢ (∀xRel A ↔ ∀x A ⊆ (V × V))
6	3, 4, 5	3imtr4 192	1 ⊢ (Rel A → ∀xRel A)

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  hbfun 2684

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-rel 2425

metamath.org