Theorem f1ocnvfvb 3878

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 3877	. . 3 |- ((F:A-1-1-onto->B /\ C e. A) -> ((F` C) = D -> (`'F` D) = C))
2	1	3adant3 798	. 2 |- ((F:A-1-1-onto->B /\ C e. A /\ D e. B) -> ((F` C) = D -> (`'F` D) = C))
3		f1ocnvfv2 3876	. . . . 5 |- ((F:A-1-1-onto->B /\ D e. B) -> (F` (`'F` D)) = D)
4	3	eqeq2d 1485	. . . 4 |- ((F:A-1-1-onto->B /\ D e. B) -> ((F` C) = (F` (`'F` D)) <-> (F` C) = D))
5		fveq2 3721	. . . . 5 |- (C = (`'F` D) -> (F` C) = (F` (`'F` D)))
6	5	eqcoms 1477	. . . 4 |- ((`'F` D) = C -> (F` C) = (F` (`'F` D)))
7	4, 6	syl5bi 208	. . 3 |- ((F:A-1-1-onto->B /\ D e. B) -> ((`'F` D) = C -> (F` C) = D))
8	7	3adant2 797	. 2 |- ((F:A-1-1-onto->B /\ C e. A /\ D e. B) -> ((`'F` D) = C -> (F` C) = D))
9	2, 8	impbid 515	1 |- ((F:A-1-1-onto->B /\ C e. A /\ D e. B) -> ((F` C) = D <-> (`'F` D) = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  `'ccnv 3166  -1-1-onto->wf1o 3178  ` cfv 3179

This theorem is referenced by:  f1ofveu 3879  logeftb 8748  bracnlnvalt 10038

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 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195

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