Theorem oooeqim2 10476

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 (_ A /\ Y (_ A) -> ((F"X) = (F"Y) <-> X = Y))

Proof of Theorem oooeqim2

Step	Hyp	Ref	Expression
1		f1imacnv 3705	. . . . . . 7 |- ((F:A-1-1->B /\ X (_ A) -> (`'F"(F"X)) = X)
2	1	ex 373	. . . . . 6 |- (F:A-1-1->B -> (X (_ A -> (`'F"(F"X)) = X))
3		f1imacnv 3705	. . . . . . 7 |- ((F:A-1-1->B /\ Y (_ A) -> (`'F"(F"Y)) = Y)
4	3	ex 373	. . . . . 6 |- (F:A-1-1->B -> (Y (_ A -> (`'F"(F"Y)) = Y))
5	2, 4	anim12d 558	. . . . 5 |- (F:A-1-1->B -> ((X (_ A /\ Y (_ A) -> ((`'F"(F"X)) = X /\ (`'F"(F"Y)) = Y)))
6	5	3impib 831	. . . 4 |- ((F:A-1-1->B /\ X (_ A /\ Y (_ A) -> ((`'F"(F"X)) = X /\ (`'F"(F"Y)) = Y))
7		eqtrt 1492	. . . . . . . . 9 |- ((X = (`'F"(F"X)) /\ (`'F"(F"X)) = (`'F"(F"Y))) -> X = (`'F"(F"Y)))
8		eqtrt 1492	. . . . . . . . . 10 |- ((X = (`'F"(F"Y)) /\ (`'F"(F"Y)) = Y) -> X = Y)
9	8	ex 373	. . . . . . . . 9 |- (X = (`'F"(F"Y)) -> ((`'F"(F"Y)) = Y -> X = Y))
10	7, 9	syl 10	. . . . . . . 8 |- ((X = (`'F"(F"X)) /\ (`'F"(F"X)) = (`'F"(F"Y))) -> ((`'F"(F"Y)) = Y -> X = Y))
11	10	ex 373	. . . . . . 7 |- (X = (`'F"(F"X)) -> ((`'F"(F"X)) = (`'F"(F"Y)) -> ((`'F"(F"Y)) = Y -> X = Y)))
12	11	com23 32	. . . . . 6 |- (X = (`'F"(F"X)) -> ((`'F"(F"Y)) = Y -> ((`'F"(F"X)) = (`'F"(F"Y)) -> X = Y)))
13	12	eqcoms 1478	. . . . 5 |- ((`'F"(F"X)) = X -> ((`'F"(F"Y)) = Y -> ((`'F"(F"X)) = (`'F"(F"Y)) -> X = Y)))
14	13	imp 350	. . . 4 |- (((`'F"(F"X)) = X /\ (`'F"(F"Y)) = Y) -> ((`'F"(F"X)) = (`'F"(F"Y)) -> X = Y))
15	6, 14	syl 10	. . 3 |- ((F:A-1-1->B /\ X (_ A /\ Y (_ A) -> ((`'F"(F"X)) = (`'F"(F"Y)) -> X = Y))
16		imaeq2 3402	. . 3 |- ((F"X) = (F"Y) -> (`'F"(F"X)) = (`'F"(F"Y)))
17	15, 16	syl5 21	. 2 |- ((F:A-1-1->B /\ X (_ A /\ Y (_ A) -> ((F"X) = (F"Y) -> X = Y))
18		imaeq2 3402	. 2 |- (X = Y -> (F"X) = (F"Y))
19	17, 18	impbid1 517	1 |- ((F:A-1-1->B /\ X (_ A /\ Y (_ A) -> ((F"X) = (F"Y) <-> X = Y))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   (_ wss 2047  `'ccnv 3169  "cima 3173  -1-1->wf1 3179

This theorem is referenced by:  homcard 10539

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-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

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-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-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

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