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Theorem elreldm 4503
Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
Assertion
Ref Expression
elreldm ((Rel A B A) → B dom A)

Proof of Theorem elreldm
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rel 4295 . . . . 5 (Rel AA ⊆ (V × V))
2 ssel 2933 . . . . 5 (A ⊆ (V × V) → (B AB (V × V)))
31, 2sylbi 114 . . . 4 (Rel A → (B AB (V × V)))
4 elvv 4345 . . . 4 (B (V × V) ↔ xy B = ⟨x, y⟩)
53, 4syl6ib 150 . . 3 (Rel A → (B Axy B = ⟨x, y⟩))
6 eleq1 2097 . . . . . 6 (B = ⟨x, y⟩ → (B A ↔ ⟨x, y A))
7 vex 2554 . . . . . . 7 x V
8 vex 2554 . . . . . . 7 y V
97, 8opeldm 4481 . . . . . 6 (⟨x, y Ax dom A)
106, 9syl6bi 152 . . . . 5 (B = ⟨x, y⟩ → (B Ax dom A))
11 inteq 3609 . . . . . . . 8 (B = ⟨x, y⟩ → B = x, y⟩)
1211inteqd 3611 . . . . . . 7 (B = ⟨x, y⟩ → B = x, y⟩)
137, 8op1stb 4175 . . . . . . 7 x, y⟩ = x
1412, 13syl6eq 2085 . . . . . 6 (B = ⟨x, y⟩ → B = x)
1514eleq1d 2103 . . . . 5 (B = ⟨x, y⟩ → ( B dom Ax dom A))
1610, 15sylibrd 158 . . . 4 (B = ⟨x, y⟩ → (B A B dom A))
1716exlimivv 1773 . . 3 (xy B = ⟨x, y⟩ → (B A B dom A))
185, 17syli 33 . 2 (Rel A → (B A B dom A))
1918imp 115 1 ((Rel A B A) → B dom A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  wss 2911  cop 3370   cint 3606   × cxp 4286  dom cdm 4288  Rel wrel 4293
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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298
This theorem is referenced by:  1stdm  5750  fundmen  6222
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