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Theorem tposfo2 6881
Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposfo2  |-  ( Rel 
A  ->  ( F : A -onto-> B  -> tpos  F : `' A -onto-> B ) )

Proof of Theorem tposfo2
StepHypRef Expression
1 tposfn2 6880 . . . 4  |-  ( Rel 
A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
21adantrd 468 . . 3  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  -> tpos  F  Fn  `' A ) )
3 fndm 5621 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
43releqd 5035 . . . . . . . 8  |-  ( F  Fn  A  ->  ( Rel  dom  F  <->  Rel  A ) )
54biimparc 487 . . . . . . 7  |-  ( ( Rel  A  /\  F  Fn  A )  ->  Rel  dom 
F )
6 rntpos 6871 . . . . . . 7  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
75, 6syl 16 . . . . . 6  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ran tpos  F  =  ran  F )
87eqeq1d 2456 . . . . 5  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ( ran tpos  F  =  B  <->  ran  F  =  B ) )
98biimprd 223 . . . 4  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ( ran  F  =  B  ->  ran tpos  F  =  B ) )
109expimpd 603 . . 3  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  ->  ran tpos  F  =  B ) )
112, 10jcad 533 . 2  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  ->  (tpos  F  Fn  `' A  /\  ran tpos  F  =  B ) ) )
12 df-fo 5535 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
13 df-fo 5535 . 2  |-  (tpos  F : `' A -onto-> B  <->  (tpos  F  Fn  `' A  /\  ran tpos  F  =  B ) )
1411, 12, 133imtr4g 270 1  |-  ( Rel 
A  ->  ( F : A -onto-> B  -> tpos  F : `' A -onto-> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   `'ccnv 4950   dom cdm 4951   ran crn 4952   Rel wrel 4956    Fn wfn 5524   -onto->wfo 5527  tpos ctpos 6857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-fo 5535  df-fv 5537  df-tpos 6858
This theorem is referenced by:  tposf2  6882  tposf1o2  6884  tposfo  6885  oppglsm  16266
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