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Theorem releldm2 5753
Description: Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
Assertion
Ref Expression
releldm2  Rel  dom  1st `
Distinct variable groups:   ,   ,

Proof of Theorem releldm2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2560 . . 3  dom  _V
21anim2i 324 . 2  Rel  dom  Rel  _V
3 id 19 . . . . 5  1st `  1st `
4 vex 2554 . . . . . 6 
_V
5 1stexg 5736 . . . . . 6  _V  1st ` 
_V
64, 5ax-mp 7 . . . . 5  1st `  _V
73, 6syl6eqelr 2126 . . . 4  1st `  _V
87rexlimivw 2423 . . 3  1st `  _V
98anim2i 324 . 2  Rel  1st `  Rel  _V
10 eldm2g 4474 . . . 4  _V  dom  <. ,  >.
1110adantl 262 . . 3  Rel  _V  dom  <. ,  >.
12 df-rel 4295 . . . . . . . . 9  Rel  C_  _V  X.  _V
13 ssel 2933 . . . . . . . . 9 
C_  _V  X.  _V  _V  X.  _V
1412, 13sylbi 114 . . . . . . . 8  Rel  _V  X.  _V
1514imp 115 . . . . . . 7  Rel  _V  X.  _V
16 op1steq 5747 . . . . . . 7  _V  X.  _V  1st `  <. ,  >.
1715, 16syl 14 . . . . . 6  Rel  1st `  <. ,  >.
1817rexbidva 2317 . . . . 5  Rel  1st `  <. , 
>.
1918adantr 261 . . . 4  Rel  _V  1st `  <. , 
>.
20 rexcom4 2571 . . . . 5  <. ,  >.  <. ,  >.
21 risset 2346 . . . . . 6  <. ,  >.  <. ,  >.
2221exbii 1493 . . . . 5  <. , 
>.  <. ,  >.
2320, 22bitr4i 176 . . . 4  <. ,  >.  <. ,  >.
2419, 23syl6bb 185 . . 3  Rel  _V  1st `  <. ,  >.
2511, 24bitr4d 180 . 2  Rel  _V  dom  1st `
262, 9, 25pm5.21nd 824 1  Rel  dom  1st `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390  wrex 2301   _Vcvv 2551    C_ wss 2911   <.cop 3370    X. cxp 4286   dom cdm 4288   Rel wrel 4293   ` 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-bnd 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:  reldm  5754
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