Theorem hbfn 2720

Description: Bound-variable hypothesis builder for a function with domain.

Hypotheses

Ref	Expression
hbfn.1	⊢ (y ∈ F → ∀x y ∈ F)
hbfn.2	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
hbfn	⊢ (F Fn A → ∀x F Fn A)

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

Proof of Theorem hbfn

Step	Hyp	Ref	Expression
1		hbfn.1	. . . 4 ⊢ (y ∈ F → ∀x y ∈ F)
2	1	hbfun 2684	. . 3 ⊢ (Fun F → ∀xFun F)
3	1	hbdm 2565	. . . 4 ⊢ (y ∈ dom F → ∀x y ∈ dom F)
4		hbfn.2	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
5	3, 4	hbeq 1171	. . 3 ⊢ (dom F = A → ∀xdom F = A)
6	2, 5	hban 704	. 2 ⊢ ((Fun F ∧ dom F = A) → ∀x(Fun F ∧ dom F = A))
7		df-fn 2433	. 2 ⊢ (F Fn A ↔ (Fun F ∧ dom F = A))
8	7	bial 695	. 2 ⊢ (∀x F Fn A ↔ ∀x(Fun F ∧ dom F = A))
9	6, 7, 8	3imtr4 192	1 ⊢ (F Fn A → ∀x F Fn A)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  dom cdm 2410  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  fnopabg 2745  hbf 2751  hbfo 2787

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-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433

metamath.org