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Theorem enf1f1oOLD 26500
Description: A one-to-one mapping of finite sets with the same cardinality is bijective. (Moved to f1finf1o 7102 in main set.mm and may be deleted by mathbox owner, JM. --NM 24-Sep-2013.) (Contributed by Jeff Madsen, 5-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
enf1f1oOLD  |-  ( ( A  e.  Fin  /\  B  ~~  A )  -> 
( F : A -1-1-> B  ->  F : A -1-1-onto-> B
) )

Proof of Theorem enf1f1oOLD
StepHypRef Expression
1 ensym 6926 . . . 4  |-  ( B 
~~  A  ->  A  ~~  B )
21adantl 452 . . 3  |-  ( ( A  e.  Fin  /\  B  ~~  A )  ->  A  ~~  B )
3 enfii 7096 . . 3  |-  ( ( A  e.  Fin  /\  B  ~~  A )  ->  B  e.  Fin )
4 f1finf1o 7102 . . 3  |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  <-> 
F : A -1-1-onto-> B ) )
52, 3, 4syl2anc 642 . 2  |-  ( ( A  e.  Fin  /\  B  ~~  A )  -> 
( F : A -1-1-> B  <-> 
F : A -1-1-onto-> B ) )
65biimpd 198 1  |-  ( ( A  e.  Fin  /\  B  ~~  A )  -> 
( F : A -1-1-> B  ->  F : A -1-1-onto-> B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   class class class wbr 4039   -1-1->wf1 5268   -1-1-onto->wf1o 5270    ~~ cen 6876   Fincfn 6879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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