Theorem f1ofveu 3882

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 3647	. . 3 |- ((`'F:B-->A /\ C e. B) -> E!x e. A <.C, x>. e. `'F)
2		f1ocnv 3701	. . . 4 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
3		f1of 3689	. . . 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 3881	. . . . . 6 |- ((F:A-1-1-onto->B /\ x e. A /\ C e. B) -> ((F` x) = C <-> (`'F` C) = x))
7	6	3com23 839	. . . . 5 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((F` x) = C <-> (`'F` C) = x))
8		visset 1813	. . . . . . . 8 |- x e. V
9	8	fnopfvb 3754	. . . . . . 7 |- ((`'F Fn B /\ C e. B) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
10	9	3adant3 799	. . . . . 6 |- ((`'F Fn B /\ C e. B /\ x e. A) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
11		f1o4 3696	. . . . . . 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 859	. . . . 5 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((`'F` C) = x <-> <.C, x>. e. `'F))
14	7, 13	bitrd 528	. . . 4 |- ((F:A-1-1-onto->B /\ C e. B /\ x e. A) -> ((F` x) = C <-> <.C, x>. e. `'F))
15	14	3expa 833	. . 3 |- (((F:A-1-1-onto->B /\ C e. B) /\ x e. A) -> ((F` x) = C <-> <.C, x>. e. `'F))
16	15	reubidva 1779	. 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 775   = wceq 956   e. wcel 958  E!wreu 1647  <.cop 2411  `'ccnv 3169   Fn wfn 3177  -->wf 3178  -1-1-onto->wf1o 3181  ` cfv 3182

This theorem is referenced by:  f1ocnvfv3 3883

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-9 965  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  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-ral 1649  df-rex 1650  df-reu 1651  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-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198

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