Theorem oooeqim2 14356

Description: Symmetrical equality of the images and of their antecedents when the mapping is one to one.

Assertion

Ref	Expression
oooeqim2	|- ((F:A-1-1->B /\ X C_ A /\ Y C_ A) -> ((F"X) = (F"Y) <-> X = Y))

Proof of Theorem oooeqim2

Step	Hyp	Ref	Expression
1		f1imacnv 4656	. . . . . . 7 |- ((F:A-1-1->B /\ X C_ A) -> (`'F"(F"X)) = X)
2	1	ex 402	. . . . . 6 |- (F:A-1-1->B -> (X C_ A -> (`'F"(F"X)) = X))
3		f1imacnv 4656	. . . . . . 7 |- ((F:A-1-1->B /\ Y C_ A) -> (`'F"(F"Y)) = Y)
4	3	ex 402	. . . . . 6 |- (F:A-1-1->B -> (Y C_ A -> (`'F"(F"Y)) = Y))
5	2, 4	anim12d 617	. . . . 5 |- (F:A-1-1->B -> ((X C_ A /\ Y C_ A) -> ((`'F"(F"X)) = X /\ (`'F"(F"Y)) = Y)))
6	5	3impib 1065	. . . 4 |- ((F:A-1-1->B /\ X C_ A /\ Y C_ A) -> ((`'F"(F"X)) = X /\ (`'F"(F"Y)) = Y))
7		eqtr 1904	. . . . . . . . 9 |- ((X = (`'F"(F"X)) /\ (`'F"(F"X)) = (`'F"(F"Y))) -> X = (`'F"(F"Y)))
8		eqtr 1904	. . . . . . . . . 10 |- ((X = (`'F"(F"Y)) /\ (`'F"(F"Y)) = Y) -> X = Y)
9	8	ex 402	. . . . . . . . 9 |- (X = (`'F"(F"Y)) -> ((`'F"(F"Y)) = Y -> X = Y))
10	7, 9	syl 12	. . . . . . . 8 |- ((X = (`'F"(F"X)) /\ (`'F"(F"X)) = (`'F"(F"Y))) -> ((`'F"(F"Y)) = Y -> X = Y))
11	10	ex 402	. . . . . . 7 |- (X = (`'F"(F"X)) -> ((`'F"(F"X)) = (`'F"(F"Y)) -> ((`'F"(F"Y)) = Y -> X = Y)))
12	11	com23 36	. . . . . 6 |- (X = (`'F"(F"X)) -> ((`'F"(F"Y)) = Y -> ((`'F"(F"X)) = (`'F"(F"Y)) -> X = Y)))
13	12	eqcoms 1887	. . . . 5 |- ((`'F"(F"X)) = X -> ((`'F"(F"Y)) = Y -> ((`'F"(F"X)) = (`'F"(F"Y)) -> X = Y)))
14	13	imp 377	. . . 4 |- (((`'F"(F"X)) = X /\ (`'F"(F"Y)) = Y) -> ((`'F"(F"X)) = (`'F"(F"Y)) -> X = Y))
15	6, 14	syl 12	. . 3 |- ((F:A-1-1->B /\ X C_ A /\ Y C_ A) -> ((`'F"(F"X)) = (`'F"(F"Y)) -> X = Y))
16		imaeq2 4260	. . 3 |- ((F"X) = (F"Y) -> (`'F"(F"X)) = (`'F"(F"Y)))
17	15, 16	syl5 20	. 2 |- ((F:A-1-1->B /\ X C_ A /\ Y C_ A) -> ((F"X) = (F"Y) -> X = Y))
18		imaeq2 4260	. 2 |- (X = Y -> (F"X) = (F"Y))
19	17, 18	impbid1 575	1 |- ((F:A-1-1->B /\ X C_ A /\ Y C_ A) -> ((F"X) = (F"Y) <-> X = Y))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   C_ wss 2593  `'ccnv 3985  "cima 3989  -1-1->wf1 3995

This theorem is referenced by:  homcard 14893

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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