Theorem f1ofv 3872

Description: A one-to-one onto function in terms of function values.

Assertion

Ref	Expression
f1ofv	|- (F:A-1-1-onto->B <-> (F Fn A /\ ran F = B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))

Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem f1ofv

Step	Hyp	Ref	Expression
1		df-f1o 3193	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
2		f1fv 3869	. . 3 |- (F:A-1-1->B <-> (F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
3		df-fo 3192	. . 3 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4	2, 3	anbi12i 482	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> ((F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)) /\ (F Fn A /\ ran F = B)))
5		df-3an 776	. . 3 |- ((F Fn A /\ ran F = B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)) <-> ((F Fn A /\ ran F = B) /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
6		eqimss 2106	. . . . . . 7 |- (ran F = B -> ran F (_ B)
7	6	anim2i 335	. . . . . 6 |- ((F Fn A /\ ran F = B) -> (F Fn A /\ ran F (_ B))
8		df-f 3190	. . . . . 6 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
9	7, 8	sylibr 200	. . . . 5 |- ((F Fn A /\ ran F = B) -> F:A-->B)
10	9	pm4.71ri 637	. . . 4 |- ((F Fn A /\ ran F = B) <-> (F:A-->B /\ (F Fn A /\ ran F = B)))
11	10	anbi1i 481	. . 3 |- (((F Fn A /\ ran F = B) /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)) <-> ((F:A-->B /\ (F Fn A /\ ran F = B)) /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
12		an23 485	. . 3 |- (((F:A-->B /\ (F Fn A /\ ran F = B)) /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)) <-> ((F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)) /\ (F Fn A /\ ran F = B)))
13	5, 11, 12	3bitrr 178	. 2 |- (((F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)) /\ (F Fn A /\ ran F = B)) <-> (F Fn A /\ ran F = B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
14	1, 4, 13	3bitr 177	1 |- (F:A-1-1-onto->B <-> (F Fn A /\ ran F = B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955  A.wral 1643   (_ wss 2044  ran crn 3167   Fn wfn 3173  -->wf 3174  -1-1->wf1 3175  -onto->wfo 3176  -1-1-onto->wf1o 3177  ` cfv 3178

This theorem is referenced by:  grpinvf 8041

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 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194

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