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Theorem euval 144
 Description: Value of the 'exists unique' predicate. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypothesis
Ref Expression
alval.1 F:(α → ∗)
Assertion
Ref Expression
euval ⊤⊧[(∃!F) = (λy:α (λx:α [(Fx:α) = [x:α = y:α]]))]
Distinct variable groups:   x,y,α   y,F,x

Proof of Theorem euval
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 weu 141 . . 3 ∃!:((α → ∗) → ∗)
2 alval.1 . . 3 F:(α → ∗)
31, 2wc 50 . 2 (∃!F):∗
4 df-eu 133 . . 3 ⊤⊧[∃! = λp:(α → ∗) (λy:α (λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]))]
51, 2, 4ceq1 89 . 2 ⊤⊧[(∃!F) = (λp:(α → ∗) (λy:α (λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]))F)]
6 wex 139 . . . 4 :((α → ∗) → ∗)
7 wal 134 . . . . . 6 :((α → ∗) → ∗)
8 wv 64 . . . . . . . . 9 p:(α → ∗):(α → ∗)
9 wv 64 . . . . . . . . 9 x:α:α
108, 9wc 50 . . . . . . . 8 (p:(α → ∗)x:α):∗
11 wv 64 . . . . . . . . 9 y:α:α
129, 11weqi 76 . . . . . . . 8 [x:α = y:α]:∗
1310, 12weqi 76 . . . . . . 7 [(p:(α → ∗)x:α) = [x:α = y:α]]:∗
1413wl 66 . . . . . 6 λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]:(α → ∗)
157, 14wc 50 . . . . 5 (λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]):∗
1615wl 66 . . . 4 λy:α (λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]):(α → ∗)
176, 16wc 50 . . 3 (λy:α (λx:α [(p:(α → ∗)x:α) = [x:α = y:α]])):∗
18 weq 41 . . . . . . . 8 = :(∗ → (∗ → ∗))
198, 2weqi 76 . . . . . . . . . 10 [p:(α → ∗) = F]:∗
2019id 25 . . . . . . . . 9 [p:(α → ∗) = F]⊧[p:(α → ∗) = F]
218, 9, 20ceq1 89 . . . . . . . 8 [p:(α → ∗) = F]⊧[(p:(α → ∗)x:α) = (Fx:α)]
2218, 10, 12, 21oveq1 99 . . . . . . 7 [p:(α → ∗) = F]⊧[[(p:(α → ∗)x:α) = [x:α = y:α]] = [(Fx:α) = [x:α = y:α]]]
2313, 22leq 91 . . . . . 6 [p:(α → ∗) = F]⊧[λx:α [(p:(α → ∗)x:α) = [x:α = y:α]] = λx:α [(Fx:α) = [x:α = y:α]]]
247, 14, 23ceq2 90 . . . . 5 [p:(α → ∗) = F]⊧[(λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]) = (λx:α [(Fx:α) = [x:α = y:α]])]
2515, 24leq 91 . . . 4 [p:(α → ∗) = F]⊧[λy:α (λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]) = λy:α (λx:α [(Fx:α) = [x:α = y:α]])]
266, 16, 25ceq2 90 . . 3 [p:(α → ∗) = F]⊧[(λy:α (λx:α [(p:(α → ∗)x:α) = [x:α = y:α]])) = (λy:α (λx:α [(Fx:α) = [x:α = y:α]]))]
2717, 2, 26cl 116 . 2 ⊤⊧[(λp:(α → ∗) (λy:α (λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]))F) = (λy:α (λx:α [(Fx:α) = [x:α = y:α]]))]
283, 5, 27eqtri 95 1 ⊤⊧[(∃!F) = (λy:α (λx:α [(Fx:α) = [x:α = y:α]]))]
 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  ∃tex 123  ∃!teu 125 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-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 This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129  df-ex 131  df-eu 133 This theorem is referenced by: (None)
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