Theorem f1ocnvfv1 3873

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

Assertion

Ref	Expression
f1ocnvfv1	|- ((F:A-1-1-onto->B /\ C e. A) -> (`'F` (F` C)) = C)

Proof of Theorem f1ocnvfv1

Step	Hyp	Ref	Expression
1		f1ococnv1 3704	. . . 4 |- (F:A-1-1-onto->B -> (`'F o. F) = (I |` A))
2	1	fveq1d 3721	. . 3 |- (F:A-1-1-onto->B -> ((`'F o. F)` C) = ((I |` A)` C))
3	2	adantr 389	. 2 |- ((F:A-1-1-onto->B /\ C e. A) -> ((`'F o. F)` C) = ((I |` A)` C))
4		fvco3 3771	. . . 4 |- ((Fun `'F /\ F:A-->B /\ C e. A) -> ((`'F o. F)` C) = (`'F` (F` C)))
5	4	3expa 832	. . 3 |- (((Fun `'F /\ F:A-->B) /\ C e. A) -> ((`'F o. F)` C) = (`'F` (F` C)))
6		f1ocnv 3696	. . . . 5 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
7		f1ofun 3686	. . . . 5 |- (`'F:B-1-1-onto->A -> Fun `'F)
8	6, 7	syl 10	. . . 4 |- (F:A-1-1-onto->B -> Fun `'F)
9		f1of 3684	. . . 4 |- (F:A-1-1-onto->B -> F:A-->B)
10	8, 9	jca 288	. . 3 |- (F:A-1-1-onto->B -> (Fun `'F /\ F:A-->B))
11	5, 10	sylan 448	. 2 |- ((F:A-1-1-onto->B /\ C e. A) -> ((`'F o. F)` C) = (`'F` (F` C)))
12		fvresi 3838	. . 3 |- (C e. A -> ((I |` A)` C) = C)
13	12	adantl 388	. 2 |- ((F:A-1-1-onto->B /\ C e. A) -> ((I |` A)` C) = C)
14	3, 11, 13	3eqtr3d 1513	1 |- ((F:A-1-1-onto->B /\ C e. A) -> (`'F` (F` C)) = C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  Icid 2827  `'ccnv 3165   |` cres 3168   o. ccom 3170  Fun wfun 3172  -->wf 3174  -1-1-onto->wf1o 3177  ` cfv 3178

This theorem is referenced by:  f1ocnvfv 3875  logeft 8717  cnvbrabrat 10001

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-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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