Theorem f1ocnvfvb 3957

Description: Relationship between the value of a one-to-one onto function and the value of its converse.

Assertion

Ref	Expression
f1ocnvfvb	|- ((F:A-1-1-onto->B /\ C e. A /\ D e. B) -> ((F` C) = D <-> (`'F` D) = C))

Proof of Theorem f1ocnvfvb

Step	Hyp	Ref	Expression
1		f1ocnvfv 3956	. . 3 |- ((F:A-1-1-onto->B /\ C e. A) -> ((F` C) = D -> (`'F` D) = C))
2	1	3adant3 802	. 2 |- ((F:A-1-1-onto->B /\ C e. A /\ D e. B) -> ((F` C) = D -> (`'F` D) = C))
3		f1ocnvfv2 3955	. . . . 5 |- ((F:A-1-1-onto->B /\ D e. B) -> (F` (`'F` D)) = D)
4	3	eqeq2d 1523	. . . 4 |- ((F:A-1-1-onto->B /\ D e. B) -> ((F` C) = (F` (`'F` D)) <-> (F` C) = D))
5		fveq2 3800	. . . . 5 |- (C = (`'F` D) -> (F` C) = (F` (`'F` D)))
6	5	eqcoms 1515	. . . 4 |- ((`'F` D) = C -> (F` C) = (F` (`'F` D)))
7	4, 6	syl5bi 206	. . 3 |- ((F:A-1-1-onto->B /\ D e. B) -> ((`'F` D) = C -> (F` C) = D))
8	7	3adant2 801	. 2 |- ((F:A-1-1-onto->B /\ C e. A /\ D e. B) -> ((`'F` D) = C -> (F` C) = D))
9	2, 8	impbid 518	1 |- ((F:A-1-1-onto->B /\ C e. A /\ D e. B) -> ((F` C) = D <-> (`'F` D) = C))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   /\ w3a 778   = wceq 988   e. wcel 990  `'ccnv 3224  -1-1-onto->wf1o 3236  ` cfv 3237

This theorem is referenced by:  f1ofveu 3958  logeftb 8883  bracnlnval 10164

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-fv 3253

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