Theorem f1ofv 3877

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 3197	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
2		f1fv 3874	. . 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 3196	. . 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 777	. . 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 2109	. . . . . . 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 3194	. . . . . 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 638	. . . 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 775   = wceq 956  A.wral 1645   (_ wss 2047  ran crn 3171   Fn wfn 3177  -->wf 3178  -1-1->wf1 3179  -onto->wfo 3180  -1-1-onto->wf1o 3181  ` cfv 3182

This theorem is referenced by:  grpinvf 8079

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-9 965  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  ax-un 2866

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-ral 1649  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-uni 2504  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  df-fv 3198

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