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Theorem dffo3f 37411
Description: An onto mapping expressed in terms of function values. As dffo3 6052 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
dffo3f.1  |-  F/_ x F
Assertion
Ref Expression
dffo3f  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
Distinct variable groups:    x, A, y    x, B, y    y, F
Allowed substitution hint:    F( x)

Proof of Theorem dffo3f
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dffo2 5814 . 2  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
2 ffn 5746 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
3 fnrnfv 5927 . . . . . . 7  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. w  e.  A  y  =  ( F `  w ) } )
4 nfcv 2580 . . . . . . . . . . 11  |-  F/_ x
y
5 dffo3f.1 . . . . . . . . . . . 12  |-  F/_ x F
6 nfcv 2580 . . . . . . . . . . . 12  |-  F/_ x w
75, 6nffv 5888 . . . . . . . . . . 11  |-  F/_ x
( F `  w
)
84, 7nfeq 2591 . . . . . . . . . 10  |-  F/ x  y  =  ( F `  w )
9 nfv 1755 . . . . . . . . . 10  |-  F/ w  y  =  ( F `  x )
10 fveq2 5881 . . . . . . . . . . 11  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
1110eqeq2d 2436 . . . . . . . . . 10  |-  ( w  =  x  ->  (
y  =  ( F `
 w )  <->  y  =  ( F `  x ) ) )
128, 9, 11cbvrex 3051 . . . . . . . . 9  |-  ( E. w  e.  A  y  =  ( F `  w )  <->  E. x  e.  A  y  =  ( F `  x ) )
1312abbii 2551 . . . . . . . 8  |-  { y  |  E. w  e.  A  y  =  ( F `  w ) }  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
1413a1i 11 . . . . . . 7  |-  ( F  Fn  A  ->  { y  |  E. w  e.  A  y  =  ( F `  w ) }  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
153, 14eqtrd 2463 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
1615eqeq1d 2424 . . . . 5  |-  ( F  Fn  A  ->  ( ran  F  =  B  <->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B ) )
172, 16syl 17 . . . 4  |-  ( F : A --> B  -> 
( ran  F  =  B 
<->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B ) )
18 nfcv 2580 . . . . . . . . . 10  |-  F/_ x A
19 nfcv 2580 . . . . . . . . . 10  |-  F/_ x B
205, 18, 19nff 5742 . . . . . . . . 9  |-  F/ x  F : A --> B
21 nfv 1755 . . . . . . . . 9  |-  F/ x  y  e.  B
22 simpr 462 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  y  =  ( F `  x ) )
23 ffvelrn 6035 . . . . . . . . . . . 12  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
2423adantr 466 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  ( F `  x )  e.  B
)
2522, 24eqeltrd 2507 . . . . . . . . . 10  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  y  e.  B )
2625exp31 607 . . . . . . . . 9  |-  ( F : A --> B  -> 
( x  e.  A  ->  ( y  =  ( F `  x )  ->  y  e.  B
) ) )
2720, 21, 26rexlimd 2906 . . . . . . . 8  |-  ( F : A --> B  -> 
( E. x  e.  A  y  =  ( F `  x )  ->  y  e.  B
) )
2827biantrurd 510 . . . . . . 7  |-  ( F : A --> B  -> 
( ( y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) )  <->  ( ( E. x  e.  A  y  =  ( F `  x )  ->  y  e.  B )  /\  (
y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) ) ) )
29 dfbi2 632 . . . . . . 7  |-  ( ( E. x  e.  A  y  =  ( F `  x )  <->  y  e.  B )  <->  ( ( E. x  e.  A  y  =  ( F `  x )  ->  y  e.  B )  /\  (
y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) ) )
3028, 29syl6rbbr 267 . . . . . 6  |-  ( F : A --> B  -> 
( ( E. x  e.  A  y  =  ( F `  x )  <-> 
y  e.  B )  <-> 
( y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) ) )
3130albidv 1761 . . . . 5  |-  ( F : A --> B  -> 
( A. y ( E. x  e.  A  y  =  ( F `  x )  <->  y  e.  B )  <->  A. y
( y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) ) )
32 abeq1 2542 . . . . 5  |-  ( { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B  <->  A. y
( E. x  e.  A  y  =  ( F `  x )  <-> 
y  e.  B ) )
33 df-ral 2776 . . . . 5  |-  ( A. y  e.  B  E. x  e.  A  y  =  ( F `  x )  <->  A. y
( y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) )
3431, 32, 333bitr4g 291 . . . 4  |-  ( F : A --> B  -> 
( { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B 
<-> 
A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
3517, 34bitrd 256 . . 3  |-  ( F : A --> B  -> 
( ran  F  =  B 
<-> 
A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
3635pm5.32i 641 . 2  |-  ( ( F : A --> B  /\  ran  F  =  B )  <-> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
371, 36bitri 252 1  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437    e. wcel 1872   {cab 2407   F/_wnfc 2566   A.wral 2771   E.wrex 2772   ran crn 4854    Fn wfn 5596   -->wf 5597   -onto->wfo 5599   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609
This theorem is referenced by:  foelrnf  37422  fompt  37428
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