Theorem f1ofveu 3877

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 3642	. . 3 |- ((`'F:B-->A /\ C e. B) -> E!x e. A <.C, x>. e. `'F)
2		f1ocnv 3696	. . . 4 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
3		f1of 3684	. . . 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 448	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> E!x e. A <.C, x>. e. `'F)
6		f1ocnvfvb 3876	. . . . . 6 |- ((F:A-1-1-onto->B /\ x e. A /\ C e. B) -> ((F` x) = C <-> (`'F` C) = x))
7	6	3com23 838	. . . . 5 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((F` x) = C <-> (`'F` C) = x))
8		visset 1810	. . . . . . . 8 |- x e. V
9	8	fnopfvb 3749	. . . . . . 7 |- ((`'F Fn B /\ C e. B) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
10	9	3adant3 798	. . . . . 6 |- ((`'F Fn B /\ C e. B /\ x e. A) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
11		f1o4 3691	. . . . . . 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 858	. . . . 5 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
14	7, 13	bitrd 527	. . . 4 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((F` x) = C <-> <.C, x>. e. `'F))
15	14	3expa 832	. . 3 |- (((F:A-1-1-onto->B /\ C e. B) /\ x e. A) -> ((F` x) = C <-> <.C, x>. e. `'F))
16	15	reubidva 1777	. 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 774   = wceq 955   e. wcel 957  E!wreu 1645  <.cop 2408  `'ccnv 3165   Fn wfn 3173  -->wf 3174  -1-1-onto->wf1o 3177  ` cfv 3178

This theorem is referenced by:  f1ocnvfv3 3878

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194

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