Theorem f1ocnvfv1 3878

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 3709	. . . 4 |- (F:A-1-1-onto->B -> (`'F o. F) = (I |` A))
2	1	fveq1d 3726	. . 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 3776	. . . 4 |- ((Fun `'F /\ F:A-->B /\ C e. A) -> ((`'F o. F)` C) = (`'F` (F` C)))
5	4	3expa 833	. . 3 |- (((Fun `'F /\ F:A-->B) /\ C e. A) -> ((`'F o. F)` C) = (`'F` (F` C)))
6		f1ocnv 3701	. . . . 5 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
7		f1ofun 3691	. . . . 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 3689	. . . 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 3843	. . 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 1515	1 |- ((F:A-1-1-onto->B /\ C e. A) -> (`'F` (F` C)) = C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Icid 2831  `'ccnv 3169   |` cres 3172   o. ccom 3174  Fun wfun 3176  -->wf 3178  -1-1-onto->wf1o 3181  ` cfv 3182

This theorem is referenced by:  f1ocnvfv 3880  logeft 8762  cnvbrabrat 10045

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-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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