Theorem f1ofveu 3888

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 3653	. . 3 |- ((`'F:B-->A /\ C e. B) -> E!x e. A <.C, x>. e. `'F)
2		f1ocnv 3707	. . . 4 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
3		f1of 3695	. . . 4 |- (`'F:B-1-1-onto->A -> `'F:B-->A)
4	2, 3	syl 10	. . 3 |- (F:A-1-1-onto->B -> `'F:B-->A)
5	1, 4	sylan 450	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> E!x e. A <.C, x>. e. `'F)
6		f1ocnvfvb 3887	. . . . . 6 |- ((F:A-1-1-onto->B /\ x e. A /\ C e. B) -> ((F` x) = C <-> (`'F` C) = x))
7	6	3com23 841	. . . . 5 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((F` x) = C <-> (`'F` C) = x))
8		visset 1816	. . . . . . . 8 |- x e. V
9	8	fnopfvb 3760	. . . . . . 7 |- ((`'F Fn B /\ C e. B) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
10	9	3adant3 801	. . . . . 6 |- ((`'F Fn B /\ C e. B /\ x e. A) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
11		f1o4 3702	. . . . . . 7 |- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))
12	11	pm3.27bi 326	. . . . . 6 |- (F:A-1-1-onto->B -> `'F Fn B)
13	10, 12	syl3an1 861	. . . . 5 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
14	7, 13	bitrd 530	. . . 4 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((F` x) = C <-> <.C, x>. e. `'F))
15	14	3expa 835	. . 3 |- (((F:A-1-1-onto->B /\ C e. B) /\ x e. A) -> ((F` x) = C <-> <.C, x>. e. `'F))
16	15	reubidva 1782	. 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 196	1 |- ((F:A-1-1-onto->B /\ C e. B) -> E!x e. A (F` x) = C)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E!wreu 1650  <.cop 2415  `'ccnv 3175   Fn wfn 3183  -->wf 3184  -1-1-onto->wf1o 3187  ` cfv 3188

This theorem is referenced by:  f1ocnvfv3 3889

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204

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