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Theorem fodomb 8895
Description: Equivalence of an onto mapping and dominance for a nonempty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)
Assertion
Ref Expression
fodomb  |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B  ~<_  A ) )
Distinct variable groups:    A, f    B, f

Proof of Theorem fodomb
StepHypRef Expression
1 fof 5777 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  -> 
f : A --> B )
2 fdm 5717 . . . . . . . . . . . 12  |-  ( f : A --> B  ->  dom  f  =  A
)
31, 2syl 16 . . . . . . . . . . 11  |-  ( f : A -onto-> B  ->  dom  f  =  A
)
43eqeq1d 2456 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  A  =  (/) ) )
5 dm0rn0 5208 . . . . . . . . . . 11  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
6 forn 5780 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
76eqeq1d 2456 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( ran  f  =  (/)  <->  B  =  (/) ) )
85, 7syl5bb 257 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  B  =  (/) ) )
94, 8bitr3d 255 . . . . . . . . 9  |-  ( f : A -onto-> B  -> 
( A  =  (/)  <->  B  =  (/) ) )
109necon3bid 2712 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( A  =/=  (/)  <->  B  =/=  (/) ) )
1110biimpac 484 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  B  =/=  (/) )
12 vex 3109 . . . . . . . . . . . 12  |-  f  e. 
_V
1312dmex 6706 . . . . . . . . . . 11  |-  dom  f  e.  _V
143, 13syl6eqelr 2551 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  A  e.  _V )
15 fornex 6742 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  e.  _V )
)
1614, 15mpcom 36 . . . . . . . . 9  |-  ( f : A -onto-> B  ->  B  e.  _V )
17 0sdomg 7639 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
1816, 17syl 16 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( (/)  ~<  B  <->  B  =/=  (/) ) )
1918adantl 464 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
2011, 19mpbird 232 . . . . . 6  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  (/)  ~<  B )
2120ex 432 . . . . 5  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  (/)  ~<  B ) )
22 fodomg 8894 . . . . . . 7  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2314, 22mpcom 36 . . . . . 6  |-  ( f : A -onto-> B  ->  B  ~<_  A )
2423a1i 11 . . . . 5  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  B  ~<_  A ) )
2521, 24jcad 531 . . . 4  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2625exlimdv 1729 . . 3  |-  ( A  =/=  (/)  ->  ( E. f  f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2726imp 427 . 2  |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  ->  ( (/) 
~<  B  /\  B  ~<_  A ) )
28 sdomdomtr 7643 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  (/)  ~<  A )
29 reldom 7515 . . . . . . 7  |-  Rel  ~<_
3029brrelex2i 5030 . . . . . 6  |-  ( B  ~<_  A  ->  A  e.  _V )
3130adantl 464 . . . . 5  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  A  e.  _V )
32 0sdomg 7639 . . . . 5  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3331, 32syl 16 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3428, 33mpbid 210 . . 3  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  A  =/=  (/) )
35 fodomr 7661 . . 3  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. f 
f : A -onto-> B
)
3634, 35jca 530 . 2  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  ( A  =/=  (/)  /\  E. f 
f : A -onto-> B
) )
3727, 36impbii 188 1  |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B  ~<_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   _Vcvv 3106   (/)c0 3783   class class class wbr 4439   dom cdm 4988   ran crn 4989   -->wf 5566   -onto->wfo 5568    ~<_ cdom 7507    ~< csdm 7508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-ac2 8834
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-recs 7034  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-card 8311  df-acn 8314  df-ac 8488
This theorem is referenced by: (None)
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