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Mirrors > Home > HOLE Home > Th. List > eta | GIF version |
Description: The eta-axiom: a function is determined by its values. |
Ref | Expression |
---|---|
eta.1 | ⊢ F:(α → β) |
Ref | Expression |
---|---|
eta | ⊢ ⊤⊧[λx:α (Fx:α) = F] |
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 |
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