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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:(αβ)
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.
StepHypRef Expression
1 ax-eta 177 . 2 ⊤⊧(λf:(αβ) [λx:α (f:(αβ)x:α) = f:(αβ)])
2 weq 41 . . . 4 = :((αβ) → ((αβ) → ∗))
3 wv 64 . . . . . 6 f:(αβ):(αβ)
4 wv 64 . . . . . 6 x:α:α
53, 4wc 50 . . . . 5 (f:(αβ)x:α):β
65wl 66 . . . 4 λx:α (f:(αβ)x:α):(αβ)
72, 6, 3wov 72 . . 3 [λx:α (f:(αβ)x:α) = f:(αβ)]:∗
8 eta.1 . . 3 F:(αβ)
93, 8weqi 76 . . . . . . 7 [f:(αβ) = F]:∗
109id 25 . . . . . 6 [f:(αβ) = F]⊧[f:(αβ) = F]
113, 4, 10ceq1 89 . . . . 5 [f:(αβ) = F]⊧[(f:(αβ)x:α) = (Fx:α)]
125, 11leq 91 . . . 4 [f:(αβ) = F]⊧[λx:α (f:(αβ)x:α) = λx:α (Fx:α)]
132, 6, 3, 12, 10oveq12 100 . . 3 [f:(αβ) = F]⊧[[λx:α (f:(αβ)x:α) = f:(αβ)] = [λx:α (Fx:α) = F]]
147, 8, 13cla4v 152 . 2 (λf:(αβ) [λx:α (f:(αβ)x:α) = f:(αβ)])⊧[λx:α (Fx:α) = F]
151, 14syl 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 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
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