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Theorem fodomr 7012
Description: There exists a mapping from a set onto any (non-empty) set that it dominates. (Contributed by NM, 23-Mar-2006.)
Assertion
Ref Expression
fodomr  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. f 
f : A -onto-> B
)
Distinct variable groups:    A, f    B, f

Proof of Theorem fodomr
Dummy variables  g 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 6869 . . . 4  |-  Rel  ~<_
21brrelex2i 4730 . . 3  |-  ( B  ~<_  A  ->  A  e.  _V )
32adantl 452 . 2  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  A  e.  _V )
41brrelexi 4729 . . . 4  |-  ( B  ~<_  A  ->  B  e.  _V )
5 0sdomg 6990 . . . . 5  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
6 n0 3464 . . . . 5  |-  ( B  =/=  (/)  <->  E. z  z  e.  B )
75, 6syl6bb 252 . . . 4  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  E. z  z  e.  B ) )
84, 7syl 15 . . 3  |-  ( B  ~<_  A  ->  ( (/)  ~<  B  <->  E. z 
z  e.  B ) )
98biimpac 472 . 2  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. z 
z  e.  B )
10 brdomi 6873 . . 3  |-  ( B  ~<_  A  ->  E. g 
g : B -1-1-> A
)
1110adantl 452 . 2  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. g 
g : B -1-1-> A
)
12 difexg 4162 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  \  ran  g )  e.  _V )
13 snex 4216 . . . . . . . . . 10  |-  { z }  e.  _V
14 xpexg 4800 . . . . . . . . . 10  |-  ( ( ( A  \  ran  g )  e.  _V  /\ 
{ z }  e.  _V )  ->  ( ( A  \  ran  g
)  X.  { z } )  e.  _V )
1512, 13, 14sylancl 643 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
( A  \  ran  g )  X.  {
z } )  e. 
_V )
16 vex 2791 . . . . . . . . . 10  |-  g  e. 
_V
1716cnvex 5209 . . . . . . . . 9  |-  `' g  e.  _V
1815, 17jctil 523 . . . . . . . 8  |-  ( A  e.  _V  ->  ( `' g  e.  _V  /\  ( ( A  \  ran  g )  X.  {
z } )  e. 
_V ) )
19 unexb 4520 . . . . . . . 8  |-  ( ( `' g  e.  _V  /\  ( ( A  \  ran  g )  X.  {
z } )  e. 
_V )  <->  ( `' g  u.  ( ( A  \  ran  g )  X.  { z } ) )  e.  _V )
2018, 19sylib 188 . . . . . . 7  |-  ( A  e.  _V  ->  ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) )  e.  _V )
21 df-f1 5260 . . . . . . . . . . . . 13  |-  ( g : B -1-1-> A  <->  ( g : B --> A  /\  Fun  `' g ) )
2221simprbi 450 . . . . . . . . . . . 12  |-  ( g : B -1-1-> A  ->  Fun  `' g )
23 vex 2791 . . . . . . . . . . . . . 14  |-  z  e. 
_V
2423fconst 5427 . . . . . . . . . . . . 13  |-  ( ( A  \  ran  g
)  X.  { z } ) : ( A  \  ran  g
) --> { z }
25 ffun 5391 . . . . . . . . . . . . 13  |-  ( ( ( A  \  ran  g )  X.  {
z } ) : ( A  \  ran  g ) --> { z }  ->  Fun  ( ( A  \  ran  g
)  X.  { z } ) )
2624, 25ax-mp 8 . . . . . . . . . . . 12  |-  Fun  (
( A  \  ran  g )  X.  {
z } )
2722, 26jctir 524 . . . . . . . . . . 11  |-  ( g : B -1-1-> A  -> 
( Fun  `' g  /\  Fun  ( ( A 
\  ran  g )  X.  { z } ) ) )
28 df-rn 4700 . . . . . . . . . . . . . 14  |-  ran  g  =  dom  `' g
2928eqcomi 2287 . . . . . . . . . . . . 13  |-  dom  `' g  =  ran  g
3023snnz 3744 . . . . . . . . . . . . . 14  |-  { z }  =/=  (/)
31 dmxp 4897 . . . . . . . . . . . . . 14  |-  ( { z }  =/=  (/)  ->  dom  ( ( A  \  ran  g )  X.  {
z } )  =  ( A  \  ran  g ) )
3230, 31ax-mp 8 . . . . . . . . . . . . 13  |-  dom  (
( A  \  ran  g )  X.  {
z } )  =  ( A  \  ran  g )
3329, 32ineq12i 3368 . . . . . . . . . . . 12  |-  ( dom  `' g  i^i  dom  (
( A  \  ran  g )  X.  {
z } ) )  =  ( ran  g  i^i  ( A  \  ran  g ) )
34 disjdif 3526 . . . . . . . . . . . 12  |-  ( ran  g  i^i  ( A 
\  ran  g )
)  =  (/)
3533, 34eqtri 2303 . . . . . . . . . . 11  |-  ( dom  `' g  i^i  dom  (
( A  \  ran  g )  X.  {
z } ) )  =  (/)
36 funun 5296 . . . . . . . . . . 11  |-  ( ( ( Fun  `' g  /\  Fun  ( ( A  \  ran  g
)  X.  { z } ) )  /\  ( dom  `' g  i^i 
dom  ( ( A 
\  ran  g )  X.  { z } ) )  =  (/) )  ->  Fun  ( `' g  u.  ( ( A  \  ran  g )  X.  {
z } ) ) )
3727, 35, 36sylancl 643 . . . . . . . . . 10  |-  ( g : B -1-1-> A  ->  Fun  ( `' g  u.  ( ( A  \  ran  g )  X.  {
z } ) ) )
3837adantl 452 . . . . . . . . 9  |-  ( ( z  e.  B  /\  g : B -1-1-> A )  ->  Fun  ( `' g  u.  ( ( A  \  ran  g )  X.  { z } ) ) )
39 dmun 4885 . . . . . . . . . . . 12  |-  dom  ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) )  =  ( dom  `' g  u.  dom  ( ( A  \  ran  g
)  X.  { z } ) )
4028uneq1i 3325 . . . . . . . . . . . 12  |-  ( ran  g  u.  dom  (
( A  \  ran  g )  X.  {
z } ) )  =  ( dom  `' g  u.  dom  ( ( A  \  ran  g
)  X.  { z } ) )
4132uneq2i 3326 . . . . . . . . . . . 12  |-  ( ran  g  u.  dom  (
( A  \  ran  g )  X.  {
z } ) )  =  ( ran  g  u.  ( A  \  ran  g ) )
4239, 40, 413eqtr2i 2309 . . . . . . . . . . 11  |-  dom  ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) )  =  ( ran  g  u.  ( A  \  ran  g ) )
43 f1f 5437 . . . . . . . . . . . . 13  |-  ( g : B -1-1-> A  -> 
g : B --> A )
44 frn 5395 . . . . . . . . . . . . 13  |-  ( g : B --> A  ->  ran  g  C_  A )
4543, 44syl 15 . . . . . . . . . . . 12  |-  ( g : B -1-1-> A  ->  ran  g  C_  A )
46 undif 3534 . . . . . . . . . . . 12  |-  ( ran  g  C_  A  <->  ( ran  g  u.  ( A  \  ran  g ) )  =  A )
4745, 46sylib 188 . . . . . . . . . . 11  |-  ( g : B -1-1-> A  -> 
( ran  g  u.  ( A  \  ran  g
) )  =  A )
4842, 47syl5eq 2327 . . . . . . . . . 10  |-  ( g : B -1-1-> A  ->  dom  ( `' g  u.  ( ( A  \  ran  g )  X.  {
z } ) )  =  A )
4948adantl 452 . . . . . . . . 9  |-  ( ( z  e.  B  /\  g : B -1-1-> A )  ->  dom  ( `' g  u.  ( ( A  \  ran  g )  X.  { z } ) )  =  A )
50 df-fn 5258 . . . . . . . . 9  |-  ( ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) )  Fn  A  <->  ( Fun  ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) )  /\  dom  ( `' g  u.  ( ( A  \  ran  g
)  X.  { z } ) )  =  A ) )
5138, 49, 50sylanbrc 645 . . . . . . . 8  |-  ( ( z  e.  B  /\  g : B -1-1-> A )  ->  ( `' g  u.  ( ( A 
\  ran  g )  X.  { z } ) )  Fn  A )
52 rnun 5089 . . . . . . . . 9  |-  ran  ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) )  =  ( ran  `' g  u.  ran  ( ( A  \  ran  g
)  X.  { z } ) )
53 dfdm4 4872 . . . . . . . . . . . 12  |-  dom  g  =  ran  `' g
54 f1dm 5441 . . . . . . . . . . . 12  |-  ( g : B -1-1-> A  ->  dom  g  =  B
)
5553, 54syl5eqr 2329 . . . . . . . . . . 11  |-  ( g : B -1-1-> A  ->  ran  `' g  =  B
)
5655uneq1d 3328 . . . . . . . . . 10  |-  ( g : B -1-1-> A  -> 
( ran  `' g  u.  ran  ( ( A 
\  ran  g )  X.  { z } ) )  =  ( B  u.  ran  ( ( A  \  ran  g
)  X.  { z } ) ) )
57 0ss 3483 . . . . . . . . . . . . . 14  |-  (/)  C_  B
58 xpeq1 4703 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  \  ran  g
)  =  (/)  ->  (
( A  \  ran  g )  X.  {
z } )  =  ( (/)  X.  { z } ) )
59 xp0r 4768 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  X. 
{ z } )  =  (/)
6058, 59syl6eq 2331 . . . . . . . . . . . . . . . . 17  |-  ( ( A  \  ran  g
)  =  (/)  ->  (
( A  \  ran  g )  X.  {
z } )  =  (/) )
6160rneqd 4906 . . . . . . . . . . . . . . . 16  |-  ( ( A  \  ran  g
)  =  (/)  ->  ran  ( ( A  \  ran  g )  X.  {
z } )  =  ran  (/) )
62 rn0 4936 . . . . . . . . . . . . . . . 16  |-  ran  (/)  =  (/)
6361, 62syl6eq 2331 . . . . . . . . . . . . . . 15  |-  ( ( A  \  ran  g
)  =  (/)  ->  ran  ( ( A  \  ran  g )  X.  {
z } )  =  (/) )
6463sseq1d 3205 . . . . . . . . . . . . . 14  |-  ( ( A  \  ran  g
)  =  (/)  ->  ( ran  ( ( A  \  ran  g )  X.  {
z } )  C_  B 
<->  (/)  C_  B ) )
6557, 64mpbiri 224 . . . . . . . . . . . . 13  |-  ( ( A  \  ran  g
)  =  (/)  ->  ran  ( ( A  \  ran  g )  X.  {
z } )  C_  B )
6665a1d 22 . . . . . . . . . . . 12  |-  ( ( A  \  ran  g
)  =  (/)  ->  (
z  e.  B  ->  ran  ( ( A  \  ran  g )  X.  {
z } )  C_  B ) )
67 rnxp 5106 . . . . . . . . . . . . . . 15  |-  ( ( A  \  ran  g
)  =/=  (/)  ->  ran  ( ( A  \  ran  g )  X.  {
z } )  =  { z } )
6867adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( A  \  ran  g )  =/=  (/)  /\  z  e.  B )  ->  ran  ( ( A  \  ran  g )  X.  {
z } )  =  { z } )
69 snssi 3759 . . . . . . . . . . . . . . 15  |-  ( z  e.  B  ->  { z }  C_  B )
7069adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( A  \  ran  g )  =/=  (/)  /\  z  e.  B )  ->  { z }  C_  B )
7168, 70eqsstrd 3212 . . . . . . . . . . . . 13  |-  ( ( ( A  \  ran  g )  =/=  (/)  /\  z  e.  B )  ->  ran  ( ( A  \  ran  g )  X.  {
z } )  C_  B )
7271ex 423 . . . . . . . . . . . 12  |-  ( ( A  \  ran  g
)  =/=  (/)  ->  (
z  e.  B  ->  ran  ( ( A  \  ran  g )  X.  {
z } )  C_  B ) )
7366, 72pm2.61ine 2522 . . . . . . . . . . 11  |-  ( z  e.  B  ->  ran  ( ( A  \  ran  g )  X.  {
z } )  C_  B )
74 ssequn2 3348 . . . . . . . . . . 11  |-  ( ran  ( ( A  \  ran  g )  X.  {
z } )  C_  B 
<->  ( B  u.  ran  ( ( A  \  ran  g )  X.  {
z } ) )  =  B )
7573, 74sylib 188 . . . . . . . . . 10  |-  ( z  e.  B  ->  ( B  u.  ran  ( ( A  \  ran  g
)  X.  { z } ) )  =  B )
7656, 75sylan9eqr 2337 . . . . . . . . 9  |-  ( ( z  e.  B  /\  g : B -1-1-> A )  ->  ( ran  `' g  u.  ran  ( ( A  \  ran  g
)  X.  { z } ) )  =  B )
7752, 76syl5eq 2327 . . . . . . . 8  |-  ( ( z  e.  B  /\  g : B -1-1-> A )  ->  ran  ( `' g  u.  ( ( A  \  ran  g )  X.  { z } ) )  =  B )
78 df-fo 5261 . . . . . . . 8  |-  ( ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) ) : A -onto-> B  <->  ( ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) )  Fn  A  /\  ran  ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) )  =  B ) )
7951, 77, 78sylanbrc 645 . . . . . . 7  |-  ( ( z  e.  B  /\  g : B -1-1-> A )  ->  ( `' g  u.  ( ( A 
\  ran  g )  X.  { z } ) ) : A -onto-> B
)
80 foeq1 5447 . . . . . . . 8  |-  ( f  =  ( `' g  u.  ( ( A 
\  ran  g )  X.  { z } ) )  ->  ( f : A -onto-> B  <->  ( `' g  u.  ( ( A 
\  ran  g )  X.  { z } ) ) : A -onto-> B
) )
8180spcegv 2869 . . . . . . 7  |-  ( ( `' g  u.  (
( A  \  ran  g )  X.  {
z } ) )  e.  _V  ->  (
( `' g  u.  ( ( A  \  ran  g )  X.  {
z } ) ) : A -onto-> B  ->  E. f  f : A -onto-> B ) )
8220, 79, 81syl2im 34 . . . . . 6  |-  ( A  e.  _V  ->  (
( z  e.  B  /\  g : B -1-1-> A
)  ->  E. f 
f : A -onto-> B
) )
8382expdimp 426 . . . . 5  |-  ( ( A  e.  _V  /\  z  e.  B )  ->  ( g : B -1-1-> A  ->  E. f  f : A -onto-> B ) )
8483exlimdv 1664 . . . 4  |-  ( ( A  e.  _V  /\  z  e.  B )  ->  ( E. g  g : B -1-1-> A  ->  E. f  f : A -onto-> B ) )
8584ex 423 . . 3  |-  ( A  e.  _V  ->  (
z  e.  B  -> 
( E. g  g : B -1-1-> A  ->  E. f  f : A -onto-> B ) ) )
8685exlimdv 1664 . 2  |-  ( A  e.  _V  ->  ( E. z  z  e.  B  ->  ( E. g 
g : B -1-1-> A  ->  E. f  f : A -onto-> B ) ) )
873, 9, 11, 86syl3c 57 1  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. f 
f : A -onto-> B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  pwdom  7013  fodomfib  7136  domwdom  7288  iunfictbso  7741  fodomb  8151  brdom3  8153  konigthlem  8190  1stcfb  17171  ovoliunnul  18866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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