Theorem f1ocnvfv3 3878

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 3874	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> (F` (`'F` C)) = C)
2		fveq2 3719	. . . . . 6 |- (x = (`'F` C) -> (F` x) = (F` (`'F` C)))
3	2	eqeq1d 1481	. . . . 5 |- (x = (`'F` C) -> ((F` x) = C <-> (F` (`'F` C)) = C))
4	3	reuuni2 2880	. . . 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 1475	. . . 4 |- (U.{x e. A | (F` x) = C} = (`'F` C) <-> (`'F` C) = U.{x e. A | (F` x) = C})
6	4, 5	syl6bb 535	. . 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 3809	. . . 4 |- ((`'F:B-->A /\ C e. B) -> (`'F` C) e. A)
8		f1ocnv 3696	. . . . 5 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
9		f1of 3684	. . . . 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 3877	. . 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 955   e. wcel 957  E!wreu 1645  {crab 1646  U.cuni 2499  `'ccnv 3165  -->wf 3174  -1-1-onto->wf1o 3177  ` cfv 3178

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-rab 1650  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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