Theorem hbfun 2684

Description: Bound-variable hypothesis builder for a function.

Hypothesis

Ref	Expression
hbfun.1	⊢ (y ∈ F → ∀x y ∈ F)

Assertion

Ref	Expression
hbfun	⊢ (Fun F → ∀xFun F)

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

Proof of Theorem hbfun

Step	Hyp	Ref	Expression
1		hbfun.1	. . . 4 ⊢ (y ∈ F → ∀x y ∈ F)
2	1	hbrel 2478	. . 3 ⊢ (Rel F → ∀xRel F)
3	1	hbcnv 2516	. . . . 5 ⊢ (y ∈ ◡F → ∀x y ∈ ◡F)
4	1, 3	hbco 2508	. . . 4 ⊢ (y ∈ (F ∘ ◡F) → ∀x y ∈ (F ∘ ◡F))
5		ax-17 925	. . . 4 ⊢ (y ∈ I → ∀x y ∈ I)
6	4, 5	hbss 1501	. . 3 ⊢ ((F ∘ ◡F) ⊆ I → ∀x(F ∘ ◡F) ⊆ I)
7	2, 6	hban 704	. 2 ⊢ ((Rel F ∧ (F ∘ ◡F) ⊆ I) → ∀x(Rel F ∧ (F ∘ ◡F) ⊆ I))
8		df-fun 2432	. 2 ⊢ (Fun F ↔ (Rel F ∧ (F ∘ ◡F) ⊆ I))
9	8	bial 695	. 2 ⊢ (∀xFun F ↔ ∀x(Rel F ∧ (F ∘ ◡F) ⊆ I))
10	7, 8, 9	3imtr4 192	1 ⊢ (Fun F → ∀xFun F)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   ∈ wcel 1092   ⊆ wss 1487  Icid 2057  ^◡ccnv 2409   ∘ ccom 2414  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  hbfn 2720  hbf1 2779

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-fun 2432

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