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Theorem foov 5589
Description: An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)
Assertion
Ref Expression
foov  F :  X.  -onto-> C  F :  X.  --> C  C  F
Distinct variable groups:   ,,,   ,,,   , C   , F,,
Allowed substitution hints:    C(,)

Proof of Theorem foov
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dffo3 5257 . 2  F :  X.  -onto-> C  F :  X.  --> C  C  X.  F `
2 fveq2 5121 . . . . . . 7  <. , 
>.  F `  F `  <. ,  >.
3 df-ov 5458 . . . . . . 7  F  F `  <. ,  >.
42, 3syl6eqr 2087 . . . . . 6  <. , 
>.  F `  F
54eqeq2d 2048 . . . . 5  <. , 
>.  F `  F
65rexxp 4423 . . . 4  X.  F `  F
76ralbii 2324 . . 3  C  X.  F `  C  F
87anbi2i 430 . 2  F :  X.  --> C  C  X.  F `  F :  X.  --> C  C  F
91, 8bitri 173 1  F :  X.  -onto-> C  F :  X.  --> C  C  F
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wceq 1242  wral 2300  wrex 2301   <.cop 3370    X. cxp 4286   -->wf 4841   -onto->wfo 4843   ` cfv 4845  (class class class)co 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-ov 5458
This theorem is referenced by: (None)
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