Theorem hbres 2577

Description: Bound-variable hypothesis builder for restriction.

Hypotheses

Ref	Expression
hbres.1	⊢ (y ∈ A → ∀x y ∈ A)
hbres.2	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
hbres	⊢ (y ∈ (A ↾ B) → ∀x y ∈ (A ↾ B))

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

Proof of Theorem hbres

Step	Hyp	Ref	Expression
1		hbres.1	. . 3 ⊢ (y ∈ A → ∀x y ∈ A)
2		hbres.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
3		ax-17 925	. . . 4 ⊢ (y ∈ V → ∀x y ∈ V)
4	2, 3	hbxp 2444	. . 3 ⊢ (y ∈ (B × V) → ∀x y ∈ (B × V))
5	1, 4	hbin 1647	. 2 ⊢ (y ∈ (A ∩ (B × V)) → ∀x y ∈ (A ∩ (B × V)))
6		df-res 2430	. . 3 ⊢ (A ↾ B) = (A ∩ (B × V))
7	6	eleq2i 1153	. 2 ⊢ (y ∈ (A ↾ B) ↔ y ∈ (A ∩ (B × V)))
8	7	bial 695	. 2 ⊢ (∀x y ∈ (A ↾ B) ↔ ∀x y ∈ (A ∩ (B × V)))
9	5, 7, 8	3imtr4 192	1 ⊢ (y ∈ (A ↾ B) → ∀x y ∈ (A ↾ B))

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486   × cxp 2408   ↾ cres 2412

This theorem is referenced by:  frsucopab 2992  unbnn 3435  inf0 3457  trcl 3489  om2uzsuc 4652

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-res 2430

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