Theorem fneu 3592

Description: There is exactly one value of a function.

Assertion

Ref	Expression
fneu	|- ((F Fn A /\ B e. A) -> E!y BFy)

Distinct variable groups:   y,F   y,B

Proof of Theorem fneu

Step	Hyp	Ref	Expression
1		eleq1 1534	. . . . 5 |- (x = B -> (x e. A <-> B e. A))
2	1	anbi2d 616	. . . 4 |- (x = B -> ((F Fn A /\ x e. A) <-> (F Fn A /\ B e. A)))
3		breq1 2622	. . . . 5 |- (x = B -> (xFy <-> BFy))
4	3	eubidv 1386	. . . 4 |- (x = B -> (E!y xFy <-> E!y BFy))
5	2, 4	imbi12d 626	. . 3 |- (x = B -> (((F Fn A /\ x e. A) -> E!y xFy) <-> ((F Fn A /\ B e. A) -> E!y BFy)))
6		fndm 3587	. . . . . . . 8 |- (F Fn A -> dom F = A)
7	6	eleq2d 1541	. . . . . . 7 |- (F Fn A -> (x e. dom F <-> x e. A))
8		visset 1813	. . . . . . . 8 |- x e. V
9	8	eldm 3307	. . . . . . 7 |- (x e. dom F <-> E.y xFy)
10	7, 9	syl5bbr 534	. . . . . 6 |- (F Fn A -> (E.y xFy <-> x e. A))
11		ax-17 971	. . . . . . . . 9 |- (Fun F -> A.yFun F)
12		hbeu1 1388	. . . . . . . . 9 |- (E!y xFy -> A.yE!y xFy)
13	11, 12	hbim 1007	. . . . . . . 8 |- ((Fun F -> E!y xFy) -> A.y(Fun F -> E!y xFy))
14		funeu 3537	. . . . . . . . 9 |- ((Fun F /\ xFy) -> E!y xFy)
15	14	expcom 374	. . . . . . . 8 |- (xFy -> (Fun F -> E!y xFy))
16	13, 15	19.23ai 1064	. . . . . . 7 |- (E.y xFy -> (Fun F -> E!y xFy))
17		fnfun 3585	. . . . . . 7 |- (F Fn A -> Fun F)
18	16, 17	syl5 21	. . . . . 6 |- (E.y xFy -> (F Fn A -> E!y xFy))
19	10, 18	syl6bir 215	. . . . 5 |- (F Fn A -> (x e. A -> (F Fn A -> E!y xFy)))
20	19	pm2.43a 66	. . . 4 |- (F Fn A -> (x e. A -> E!y xFy))
21	20	imp 350	. . 3 |- ((F Fn A /\ x e. A) -> E!y xFy)
22	5, 21	vtoclg 1847	. 2 |- (B e. A -> ((F Fn A /\ B e. A) -> E!y BFy))
23	22	anabsi7 497	1 |- ((F Fn A /\ B e. A) -> E!y BFy)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  E!weu 1380   class class class wbr 2619  dom cdm 3170  Fun wfun 3176   Fn wfn 3177

This theorem is referenced by:  fneu2 3593  fnbrfvb 3753  mapsn 4345

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-cnv 3186  df-co 3187  df-dm 3188  df-fun 3192  df-fn 3193

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