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:(al -> be)

Assertion

Ref	Expression
eta	|- T. |= [\x:al (Fx:al) = F]

Distinct variable groups:   x,F   al,x   be,x

Proof of Theorem eta

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-eta 177	. 2 |- T. |= (A.\f:(al -> be) [\x:al (f:(al -> be)x:al) = f:(al -> be)])
2		weq 41	. . . 4 |- = :((al -> be) -> ((al -> be) -> *))
3		wv 64	. . . . . 6 |- f:(al -> be):(al -> be)
4		wv 64	. . . . . 6 |- x:al:al
5	3, 4	wc 50	. . . . 5 |- (f:(al -> be)x:al):be
6	5	wl 66	. . . 4 |- \x:al (f:(al -> be)x:al):(al -> be)
7	2, 6, 3	wov 72	. . 3 |- [\x:al (f:(al -> be)x:al) = f:(al -> be)]:*
8		eta.1	. . 3 |- F:(al -> be)
9	3, 8	weqi 76	. . . . . . 7 |- [f:(al -> be) = F]:*
10	9	id 25	. . . . . 6 |- [f:(al -> be) = F] |= [f:(al -> be) = F]
11	3, 4, 10	ceq1 89	. . . . 5 |- [f:(al -> be) = F] |= [(f:(al -> be)x:al) = (Fx:al)]
12	5, 11	leq 91	. . . 4 |- [f:(al -> be) = F] |= [\x:al (f:(al -> be)x:al) = \x:al (Fx:al)]
13	2, 6, 3, 12, 10	oveq12 100	. . 3 |- [f:(al -> be) = F] |= [[\x:al (f:(al -> be)x:al) = f:(al -> be)] = [\x:al (Fx:al) = F]]
14	7, 8, 13	cla4v 152	. 2 |- (A.\f:(al -> be) [\x:al (f:(al -> be)x:al) = f:(al -> be)]) |= [\x:al (Fx:al) = F]
15	1, 14	syl 16	1 |- T. |= [\x:al (Fx:al) = F]

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  A.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