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.

Step	Hyp	Ref	Expression
1		weu 141	. . 3 ⊢ ∃!:((α → ∗) → ∗)
2		alval.1	. . 3 ⊢ F:(α → ∗)
3	1, 2	wc 50	. 2 ⊢ (∃!F):∗
4		df-eu 133	. . 3 ⊢ ⊤⊧[∃! = λp:(α → ∗) (∃λy:α (∀λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]))]
5	1, 2, 4	ceq1 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
10	8, 9	wc 50	. . . . . . . 8 ⊢ (p:(α → ∗)x:α):∗
11		wv 64	. . . . . . . . 9 ⊢ y:α:α
12	9, 11	weqi 76	. . . . . . . 8 ⊢ [x:α = y:α]:∗
13	10, 12	weqi 76	. . . . . . 7 ⊢ [(p:(α → ∗)x:α) = [x:α = y:α]]:∗
14	13	wl 66	. . . . . 6 ⊢ λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]:(α → ∗)
15	7, 14	wc 50	. . . . 5 ⊢ (∀λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]):∗
16	15	wl 66	. . . 4 ⊢ λy:α (∀λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]):(α → ∗)
17	6, 16	wc 50	. . 3 ⊢ (∃λy:α (∀λx:α [(p:(α → ∗)x:α) = [x:α = y:α]])):∗
18		weq 41	. . . . . . . 8 ⊢ = :(∗ → (∗ → ∗))
19	8, 2	weqi 76	. . . . . . . . . 10 ⊢ [p:(α → ∗) = F]:∗
20	19	id 25	. . . . . . . . 9 ⊢ [p:(α → ∗) = F]⊧[p:(α → ∗) = F]
21	8, 9, 20	ceq1 89	. . . . . . . 8 ⊢ [p:(α → ∗) = F]⊧[(p:(α → ∗)x:α) = (Fx:α)]
22	18, 10, 12, 21	oveq1 99	. . . . . . 7 ⊢ [p:(α → ∗) = F]⊧[[(p:(α → ∗)x:α) = [x:α = y:α]] = [(Fx:α) = [x:α = y:α]]]
23	13, 22	leq 91	. . . . . 6 ⊢ [p:(α → ∗) = F]⊧[λx:α [(p:(α → ∗)x:α) = [x:α = y:α]] = λx:α [(Fx:α) = [x:α = y:α]]]
24	7, 14, 23	ceq2 90	. . . . 5 ⊢ [p:(α → ∗) = F]⊧[(∀λx:α [(p:(α → ∗)x:α) = [x:α = y:α]]) = (∀λx:α [(Fx:α) = [x:α = y:α]])]
25	15, 24	leq 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]])]
26	6, 16, 25	ceq2 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]]))]
27	17, 2, 26	cl 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]]))]
28	3, 5, 27	eqtri 95	1 ⊢ ⊤⊧[(∃!F) = (∃λy:α (∀λx:α [(Fx:α) = [x:α = y:α]]))]

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)