eta - Higher-Order Logic Explorer

	Higher-Order Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HOLE Home > Th. List > eta		GIF version

Theorem eta 166

Description: The eta-axiom: a function is determined by its values.

**Hypothesis**
Ref	Expression
eta.1	⊢ F:(α → β)

**Assertion**
Ref	Expression
eta	⊢ ⊤⊧[λx:α (Fx:α) = F]

Distinct variable groups: x,F α,x β,x

Proof of Theorem eta Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-eta 165	. 2 ⊢ ⊤⊧(∀λf:(α → β) [λx:α (f:(α → β)x:α) = f:(α → β)])
2		weq 38	. . . 4 ⊢ = :((α → β) → ((α → β) → ∗))
3		wv 58	. . . . . 6 ⊢ f:(α → β):(α → β)
4		wv 58	. . . . . 6 ⊢ x:α:α
5	3, 4	wc 45	. . . . 5 ⊢ (f:(α → β)x:α):β
6	5	wl 59	. . . 4 ⊢ λx:α (f:(α → β)x:α):(α → β)
7	2, 6, 3	wov 64	. . 3 ⊢ [λx:α (f:(α → β)x:α) = f:(α → β)]:∗
8		eta.1	. . 3 ⊢ F:(α → β)
9	3, 8	weqi 68	. . . . . . 7 ⊢ [f:(α → β) = F]:∗
10	9	id 25	. . . . . 6 ⊢ [f:(α → β) = F]⊧[f:(α → β) = F]
11	3, 4, 10	ceq1 79	. . . . 5 ⊢ [f:(α → β) = F]⊧[(f:(α → β)x:α) = (Fx:α)]
12	5, 11	leq 81	. . . 4 ⊢ [f:(α → β) = F]⊧[λx:α (f:(α → β)x:α) = λx:α (Fx:α)]
13	2, 6, 3, 12, 10	oveq12 90	. . 3 ⊢ [f:(α → β) = F]⊧[[λx:α (f:(α → β)x:α) = f:(α → β)] = [λx:α (Fx:α) = F]]
14	7, 8, 13	cla4v 142	. 2 ⊢ (∀λf:(α → β) [λx:α (f:(α → β)x:α) = f:(α → β)])⊧[λx:α (Fx:α) = F]
15	1, 14	syl 16	1 ⊢ ⊤⊧[λx:α (Fx:α) = F]

Colors of variables: type var term

Syntax hints: tv 1 → ht 2 ∗hb 3 kc 5 λkl 6 = ke 7 ⊤kt 8 [kbr 9 ⊧wffMMJ2 11 wffMMJ2t 12 ∀tal 112

This theorem was proved from axioms: ax-syl 15 ax-jca 17 ax-simpl 20 ax-simpr 21 ax-id 24 ax-trud 26 ax-cb1 29 ax-cb2 30 ax-refl 39 ax-eqmp 42 ax-ded 43 ax-ceq 46 ax-beta 60 ax-distrc 61 ax-leq 62 ax-hbl1 93 ax-17 95 ax-inst 103 ax-eta 165

This theorem depends on definitions: df-ov 65 df-al 116

This theorem is referenced by: cbvf 167 leqf 169 ax11 201 axext 206