Theorem f1ofv 3862

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 3187	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
2		f1fv 3859	. . 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 3186	. . 3 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4	2, 3	anbi12i 481	. 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 775	. . 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 2099	. . . . . . 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 3184	. . . . . 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 636	. . . 4 |- ((F Fn A /\ ran F = B) <-> (F:A-->B /\ (F Fn A /\ ran F = B)))
11	10	anbi1i 480	. . 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 484	. . 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 773   = wceq 953  A.wral 1637   (_ wss 2037  ran crn 3161   Fn wfn 3167  -->wf 3168  -1-1->wf1 3169  -onto->wfo 3170  -1-1-onto->wf1o 3171  ` cfv 3172

This theorem is referenced by:  grpinvf 8014

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188

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