Theorem eta 178

Description: The eta-axiom: a function is determined by its values. (Contributed by Mario Carneiro, 8-Oct-2014.)

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 177	. 2 ⊢ ⊤⊧(∀λf:(α → β) [λx:α (f:(α → β)x:α) = f:(α → β)])
2		weq 41	. . . 4 ⊢ = :((α → β) → ((α → β) → ∗))
3		wv 64	. . . . . 6 ⊢ f:(α → β):(α → β)
4		wv 64	. . . . . 6 ⊢ x:α:α
5	3, 4	wc 50	. . . . 5 ⊢ (f:(α → β)x:α):β
6	5	wl 66	. . . 4 ⊢ λx:α (f:(α → β)x:α):(α → β)
7	2, 6, 3	wov 72	. . 3 ⊢ [λx:α (f:(α → β)x:α) = f:(α → β)]:∗
8		eta.1	. . 3 ⊢ F:(α → β)
9	3, 8	weqi 76	. . . . . . 7 ⊢ [f:(α → β) = F]:∗
10	9	id 25	. . . . . 6 ⊢ [f:(α → β) = F]⊧[f:(α → β) = F]
11	3, 4, 10	ceq1 89	. . . . 5 ⊢ [f:(α → β) = F]⊧[(f:(α → β)x:α) = (Fx:α)]
12	5, 11	leq 91	. . . 4 ⊢ [f:(α → β) = F]⊧[λx:α (f:(α → β)x:α) = λx:α (Fx:α)]
13	2, 6, 3, 12, 10	oveq12 100	. . 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 152	. 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]

Syntax hints:  tv 1   → ht 2  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12  ∀tal 122

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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113  ax-eta 177

This theorem depends on definitions:  df-ov 73  df-al 126

This theorem is referenced by:  cbvf  179  leqf  181  ax11  214  axext  219