Theorem f1ocnvfv2 2920

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

Assertion

Ref	Expression
f1ocnvfv2	|- ((F:A-1-1-onto->B /\ C e. B) -> (F` (`'F` C)) = C)

Proof of Theorem f1ocnvfv2

Step	Hyp	Ref	Expression
1		f1ococnv2 2817	. . . 4 |- (F:A-1-1-onto->B -> (F o. `'F) = (I |` B))
2	1	fveq1d 2834	. . 3 |- (F:A-1-1-onto->B -> ((F o. `'F)` C) = ((I |` B)` C))
3	2	adantr 306	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> ((F o. `'F)` C) = ((I |` B)` C))
4		fvco3 2867	. . 3 |- (((Fun F /\ `'F:B-->A) /\ C e. B) -> ((F o. `'F)` C) = (F` (`'F` C)))
5		f1ofun 2802	. . . 4 |- (F:A-1-1-onto->B -> Fun F)
6		f1ocnv 2811	. . . . 5 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
7		f1of 2800	. . . . 5 |- (`'F:B-1-1-onto->A -> `'F:B-->A)
8	6, 7	syl 12	. . . 4 |- (F:A-1-1-onto->B -> `'F:B-->A)
9	5, 8	jca 236	. . 3 |- (F:A-1-1-onto->B -> (Fun F /\ `'F:B-->A))
10	4, 9	sylan 343	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> ((F o. `'F)` C) = (F` (`'F` C)))
11		fvresi 2901	. . 3 |- (C e. B -> ((I |` B)` C) = C)
12	11	adantl 305	. 2 |- ((F:A-1-1-onto->B /\ C e. B) -> ((I |` B)` C) = C)
13	3, 10, 12	3eqtr3d 1133	1 |- ((F:A-1-1-onto->B /\ C e. B) -> (F` (`'F` C)) = C)

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  Icid 2057  `'ccnv 2409   |` cres 2412   o. ccom 2414  Fun wfun 2416  -->wf 2418  -1-1-onto->wf1o 2421  ` cfv 2422

This theorem is referenced by:  f1ocnvfvb 2922  isocnv 2934  uzrdgsuc 4659

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438

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