Theorem hbrn 2564

Description: Bound-variable hypothesis builder for range.

Hypothesis

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

Assertion

Ref	Expression
hbrn	⊢ (y ∈ ran A → ∀x y ∈ ran A)

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

Proof of Theorem hbrn

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 ⊢ (w ∈ ⟨z, y⟩ → ∀x w ∈ ⟨z, y⟩)
2		ax-17 925	. . . . 5 ⊢ (y ∈ z → ∀x y ∈ z)
3		hbrn.1	. . . . 5 ⊢ (y ∈ A → ∀x y ∈ A)
4	2, 3	hbel 1172	. . . 4 ⊢ (z ∈ A → ∀x z ∈ A)
5	1, 4	hbel 1172	. . 3 ⊢ (⟨z, y⟩ ∈ A → ∀x⟨z, y⟩ ∈ A)
6	5	hbex 701	. 2 ⊢ (∃z⟨z, y⟩ ∈ A → ∀x∃z⟨z, y⟩ ∈ A)
7		visset 1350	. . 3 ⊢ y ∈ V
8	7	elrn 2562	. 2 ⊢ (y ∈ ran A ↔ ∃z⟨z, y⟩ ∈ A)
9	8	bial 695	. 2 ⊢ (∀x y ∈ ran A ↔ ∀x∃z⟨z, y⟩ ∈ A)
10	6, 8, 9	3imtr4 192	1 ⊢ (y ∈ ran A → ∀x y ∈ ran A)

Syntax hints:   → wi 2  ∀wal 672  ∃wex 678   ∈ wel 803   ∈ wcel 1092  ⟨cop 1810  ran crn 2411

This theorem is referenced by:  hbdm 2565  zfrep6 2744  hbf 2751  hbfo 2787  fopab2 2891  ac6lem 3575

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-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429

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