Theorem f1ocnvfv3 3883

Description: Value of the converse of a one-to-one onto function.

Assertion

Ref	Expression
f1ocnvfv3	|- ((F:A-1-1-onto->B /\ C e. B) -> (`'F` C) = U.{x e. A | (F` x) = C})

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

Proof of Theorem f1ocnvfv3

Step	Hyp	Ref	Expression
1		f1ocnvfv2 3879	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> (F` (`'F` C)) = C)
2		fveq2 3724	. . . . . 6 |- (x = (`'F` C) -> (F` x) = (F` (`'F` C)))
3	2	eqeq1d 1483	. . . . 5 |- (x = (`'F` C) -> ((F` x) = C <-> (F` (`'F` C)) = C))
4	3	reuuni2 2884	. . . 4 |- (((`'F` C) e. A /\ E!x e. A (F` x) = C) -> ((F` (`'F` C)) = C <-> U.{x e. A | (F` x) = C} = (`'F` C)))
5		eqcom 1477	. . . 4 |- (U.{x e. A | (F` x) = C} = (`'F` C) <-> (`'F` C) = U.{x e. A | (F` x) = C})
6	4, 5	syl6bb 536	. . 3 |- (((`'F` C) e. A /\ E!x e. A (F` x) = C) -> ((F` (`'F` C)) = C <-> (`'F` C) = U.{x e. A | (F` x) = C}))
7		ffvelrn 3814	. . . 4 |- ((`'F:B-->A /\ C e. B) -> (`'F` C) e. A)
8		f1ocnv 3701	. . . . 5 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
9		f1of 3689	. . . . 5 |- (`'F:B-1-1-onto->A -> `'F:B-->A)
10	8, 9	syl 10	. . . 4 |- (F:A-1-1-onto->B -> `'F:B-->A)
11	7, 10	sylan 448	. . 3 |- ((F:A-1-1-onto->B /\ C e. B) -> (`'F` C) e. A)
12		f1ofveu 3882	. . 3 |- ((F:A-1-1-onto->B /\ C e. B) -> E!x e. A (F` x) = C)
13	6, 11, 12	sylanc 471	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> ((F` (`'F` C)) = C <-> (`'F` C) = U.{x e. A | (F` x) = C}))
14	1, 13	mpbid 195	1 |- ((F:A-1-1-onto->B /\ C e. B) -> (`'F` C) = U.{x e. A | (F` x) = C})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E!wreu 1647  {crab 1648  U.cuni 2503  `'ccnv 3169  -->wf 3178  -1-1-onto->wf1o 3181  ` cfv 3182

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-rab 1652  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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