Theorem f1ofveu 4858

Description: There is one domain element for each value of a one-to-one onto function.

Assertion

Ref	Expression
f1ofveu	|- ((F:A-1-1-onto->B /\ C e. B) -> E!x e. A (F` x) = C)

Distinct variable groups:   x,A   x,B   x,C   x,F

Proof of Theorem f1ofveu

Step	Hyp	Ref	Expression
1		feu 4588	. . 3 |- ((`'F:B-->A /\ C e. B) -> E!x e. A <.C, x>. e. `'F)
2		f1ocnv 4651	. . . 4 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
3		f1of 4635	. . . 4 |- (`'F:B-1-1-onto->A -> `'F:B-->A)
4	2, 3	syl 12	. . 3 |- (F:A-1-1-onto->B -> `'F:B-->A)
5	1, 4	sylan 497	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> E!x e. A <.C, x>. e. `'F)
6		f1ocnvfvb 4857	. . . . . 6 |- ((F:A-1-1-onto->B /\ x e. A /\ C e. B) -> ((F` x) = C <-> (`'F` C) = x))
7	6	3com23 1074	. . . . 5 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((F` x) = C <-> (`'F` C) = x))
8		visset 2295	. . . . . . . 8 |- x e. _V
9	8	fnopfvb 4713	. . . . . . 7 |- ((`'F Fn B /\ C e. B) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
10	9	3adant3 896	. . . . . 6 |- ((`'F Fn B /\ C e. B /\ x e. A) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
11		dff1o4 4644	. . . . . . 7 |- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))
12	11	simprbi 353	. . . . . 6 |- (F:A-1-1-onto->B -> `'F Fn B)
13	10, 12	syl3an1 1130	. . . . 5 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
14	7, 13	bitrd 587	. . . 4 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((F` x) = C <-> <.C, x>. e. `'F))
15	14	3expa 1067	. . 3 |- (((F:A-1-1-onto->B /\ C e. B) /\ x e. A) -> ((F` x) = C <-> <.C, x>. e. `'F))
16	15	reubidva 2259	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> (E!x e. A (F` x) = C <-> E!x e. A <.C, x>. e. `'F))
17	5, 16	mpbird 213	1 |- ((F:A-1-1-onto->B /\ C e. B) -> E!x e. A (F` x) = C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E!wreu 2107  <.cop 3046  `'ccnv 3985   Fn wfn 3993  -->wf 3994  -1-1-onto->wf1o 3997  ` cfv 3998

This theorem is referenced by:  f1ocnvfv3 4859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014

Copyright terms: Public domain