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Theorem reldm 5754
Description: An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
Assertion
Ref Expression
reldm  Rel  dom  ran  |->  1st `
Distinct variable group:   ,

Proof of Theorem reldm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 releldm2 5753 . . 3  Rel 
dom  1st `
2 vex 2554 . . . . . . 7 
_V
3 1stexg 5736 . . . . . . 7  _V  1st ` 
_V
42, 3ax-mp 7 . . . . . 6  1st `  _V
5 eqid 2037 . . . . . 6  |->  1st `  |->  1st `
64, 5fnmpti 4970 . . . . 5  |->  1st `  Fn
7 fvelrnb 5164 . . . . 5  |->  1st `  Fn  ran  |->  1st `  |->  1st `  `
86, 7ax-mp 7 . . . 4  ran  |->  1st `  |->  1st `  `
9 fveq2 5121 . . . . . . . 8  1st `  1st `
10 vex 2554 . . . . . . . . 9 
_V
11 1stexg 5736 . . . . . . . . 9  _V  1st ` 
_V
1210, 11ax-mp 7 . . . . . . . 8  1st `  _V
139, 5, 12fvmpt 5192 . . . . . . 7  |->  1st `  `  1st `
1413eqeq1d 2045 . . . . . 6  |->  1st `  `  1st `
1514rexbiia 2333 . . . . 5  |->  1st `  `  1st `
1615a1i 9 . . . 4  Rel  |->  1st `  `  1st `
178, 16syl5rbb 182 . . 3  Rel  1st ` 
ran  |->  1st `
181, 17bitrd 177 . 2  Rel 
dom  ran  |->  1st `
1918eqrdv 2035 1  Rel  dom  ran  |->  1st `
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242   wcel 1390  wrex 2301   _Vcvv 2551    |-> cmpt 3809   dom cdm 4288   ran crn 4289   Rel wrel 4293    Fn wfn 4840   ` cfv 4845   1stc1st 5707
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-bndl 1396  ax-4 1397  ax-13 1401  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  ax-un 4136
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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  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-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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